Linear dynamical systems on Hilbert spaces: typical properties and explicit examples
Sophie Grivaux, Etienne Matheron, Quentin Menet
TL;DR
This paper explores the typical properties and explicit examples of linear operators on Hilbert spaces, revealing complex dynamical behaviors such as chaos, ergodicity, and hypercyclicity through theoretical and constructive methods.
Contribution
It provides new insights into the generic dynamical properties of Hilbert space operators and constructs explicit examples illustrating these phenomena.
Findings
Typical hypercyclic operators are not topologically mixing and have no eigenvalues.
A typical upper-triangular operator is ergodic in the Gaussian sense.
Existence of operators with complex combinations of chaotic, hypercyclic, and mixing properties.
Abstract
We solve a number of questions pertaining to the dynamics of linear operators on Hilbert spaces, sometimes by using Baire category arguments and sometimes by constructing explicit examples. In particular, we prove the following results. - A typical hypercyclic operator is not topologically mixing, has no eigenvalues and admits no non-trivial invariant measure, but is densely distributionally chaotic. - A typical upper-triangular operator is ergodic in the Gaussian sense, whereas a typical operator of the form "diagonal plus backward unilateral weighted shift" is ergodic but has only countably many unimodular eigenvalues, in particular, it is ergodic but not ergodic in the Gaussian sense. - There exist Hilbert space operators which are chaotic and -frequently hypercyclic but not frequently hypercyclic, Hilbert space operators which are chaotic and frequently hypercyclic…
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††thanks: The work of the first-named author is supported in part by EU IRSES grant AOS (PIRSES-GA-2012-318910).
Linear dynamical
systems on Hilbert spaces: typical properties and explicit examples
S. Grivaux
CNRS, Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, UMR 7352
Université de Picardie Jules Verne
33, rue Saint Leu
80039 Amiens Cedex 1
France
É. Matheron
Laboratoire de Mathématiques de Lens
Université d’Artois
Rue Jean Souvraz SP 18
62307 Lens
France
Q. Menet
Laboratoire de Mathématiques de Lens
Université d’Artois
Rue Jean Souvraz SP 18
62307 Lens
France
Résumé
We solve a number of questions pertaining to the dynamics of linear operators on Hilbert spaces, sometimes by using Baire category arguments and sometimes by constructing explicit examples. In particular, we prove the following results.
- (i)
A typical hypercyclic operator is not topologically mixing, has no eigenvalues and admits no non-trivial invariant measure, but is densely distributionally chaotic. 2. (ii)
A typical upper-triangular operator is ergodic in the Gaussian sense, whereas a typical operator of the form “diagonal plus backward unilateral weighted shift” is ergodic but has only countably many unimodular eigenvalues; in particular, it is ergodic but not ergodic in the Gaussian sense. 3. (iii)
There exist Hilbert space operators which are chaotic and -frequently hypercyclic but not frequently hypercyclic, Hilbert space operators which are chaotic and frequently hypercyclic but not ergodic, and Hilbert space operators which are chaotic and topologically mixing but not -frequently hypercyclic.
We complement our results by investigating the descriptive complexity of some natural classes of operators defined by dynamical properties.
keywords:
Linear dynamical systems; Hilbert spaces; Baire category; frequent and -frequent hypercyclicity; ergodicity; chaos; topological mixing.
1991 Mathematics Subject Classification:
47A16, 37A05, 54E52, 54H05
Table des matières
-
5.3.1 Uniform recurrence, almost periodic points and -frequent hypercyclicity
-
6.6 Infinite direct sums of frequently hypercyclic operators
-
6.8 Operators of C2-type: chaos plus mixing do not imply UFHC
1. Introduction
1.1. General overview
This paper is a contribution to the study of linear dynamical systems on Hilbert spaces. In other words, we are interested in the behavior of orbits of bounded linear operators defined on a Hilbert space. The symbol will always designate a complex separable infinite-dimensional Hilbert space, and we denote by the algebra of all bounded linear operators on .
We refer to the books [7] or [31] for background on linear dynamics, and to the papers [25] or [28] for a glimpse at the richness of the class of linear dynamical systems and its potential usefulness in general ergodic theory. A quick review of a number of definitions will be given in the next subsection. Let us just recall here that an operator is said to be hypercyclic if it admits at least one dense orbit (and hence a dense set of such orbits). The class of all hypercyclic operators on will be denoted by . Recall also that an operator is said to be chaotic if it is hypercyclic and admits a dense set of periodic vectors, and frequently hypercyclic (resp. -frequently hypercyclic) if there exists at least one vector whose orbit visits frequently every non-empty open set , in the sense that the set of integers such that belongs to has positive lower (resp. upper) density.
Obviously, the dynamical properties of linear operators can be studied (and have been studied extensively) in a very general setting – arbitrary Banach spaces, or even arbitrary topological vector spaces. However, there are good reasons for focusing on Hilbert spaces. The first that may come to mind is perhaps that “this is the natural setting for doing operator theory” – which is of course a highly questionable statement. Less subjectively, the richness of the Hilbert space structure allows for the construction of many interesting examples. Also, in some parts of linear dynamics, especially all that concerns ergodic-theoretic properties of linear operators, the general picture is neater on Hilbert spaces than on arbitrary Banach spaces. Finally, some natural questions in the area have been solved recently for operators on general Banach space but not in the Hilbertian case. This should not seem paradoxical, if one compares for example with the current state of affairs regarding the famous Invariant Subspace Problem.
The questions we are considering in this paper are quite basic. In very general terms, they are of the following type: given two interesting dynamical properties, do there exist linear operators on satisfying one of them but not the other?
Perhaps the most famous question of this type is Herrero’s “ problem”, stated in [35] and asking whether there exist hypercyclic operators which are not topologically weakly mixing. As should be expected since this is trivially true for dynamical systems on compact spaces, the answer is “Yes”: this was shown by De La Rosa and Read in [18], and then in [6] for operators on spaces, in particular for Hilbert space operators. However, the proof is surprisingly non-trivial. To give just one more example, the third named author was recently able to solve another well-known problem in the area by constructing in [39] operators on spaces which are chaotic but not frequently hypercyclic – actually, not even -frequently hypercyclic. In the present paper, we will be especially interested in the following questions:
are there operators which are frequently hypercyclic but not ergodic, i.e. do not admit an ergodic measure with full support?
- -
are there operators which are -frequently hypercyclic but not frequently hypercyclic?
- -
are there operators which are ergodic but do not admit a Gaussian ergodic measure with full support?
Note that the answer to the first two questions is known to be “Yes” for operators on the Banach space (see [9] and [29], as well as [14] for further developments); but the Hilbertian case had not been settled. As for the third question, we are not aware of any previous answer, on any Banach space.
One can attack questions of this type from two complementary points of view (of course, there are other strategies as well).
“Collective” point of view: among the properties one is interested in, one may try to determine which ones are generic and which ones are not, in a Baire category sense. Once this is done, one may be able to distinguish two properties because one of them is generic and the other one is not.
- -
“Individual” point of view: when it is unclear how a Baire category approach could work, one may still try to construct explicit examples of operators satisfying (or not) such or such a property.
Note that it is quite natural to present the two viewpoints in this order, because the indirect, Baire category approach is in some sense simpler. Indeed, it is usually not too difficult to guess when it should work, and in this case the technical details are likely to be rather soft. On the other hand, one may reasonably expect to face technical difficulties when attempting a direct construction (and perhaps quite serious ones if Baire category is unefficient).
To give meaning to the “collective” viewpoint, one has to fix an appropriate topological setting. We will in fact consider two natural topologies on : the Strong Operator Topology (denoted by SOT), and the Strong Operator Topology (denoted by ). Restricted to any closed ball , these topologies are Polish (separable and completely metrizable). Moreover, is easily seen to be SOT- in , so that for any , the family is itself a Polish space in its own right with respect to both SOT and . This opens the way to Baire category arguments in or . Of course, there is nothing new in this observation: Baire category methods have already proved as useful in operator theory as anywhere else; see in particular [21].
Recall that a subset of a Polish space is said to be meager in if it can be covered by countably many closed sets with empty interior, and comeager if its complement is meager (equivalently, if it contains a dense subset of ). Following a well-established terminology, we will say that a property of elements of is typical if the set of all satisfying it is comeager in , and that a property is atypical if its negation is typical. In this paper, we obtain among other things the following results, for any .
An -typical is (topologically) weakly mixing but not mixing.
- -
An -typical is densely distributionally chaotic.
- -
An -typical has no eigenvalues and admits no non-trivial invariant measure. In particular, chaoticity and -frequent hypercyclicity are atypical properties of hypercyclic operators (in the sense).
Admittedly, there results are not that surprising (except, perhaps, the one concerning -frequent hypercyclicity). Still, they do sketch the landscape, and they explain in some sense why some results in the area turn out to be (or are likely to be) harder to prove than some others. For example, the fact that topological weak mixing is a typical property seems to prevent the use of “soft” arguments for establishing the existence of hypercyclic operators which are not weakly mixing; and indeed, the proofs in [18] and [6] rely on rather technical arguments.
On the other hand, something more unexpected happens if one restricts oneself to the class of operators of the form , where is a diagonal operator and is a weighted unilateral backward shift (with respect to some fixed orthonormal basis of ):
in the SOT sense, most “diagonal plus shift” hypercyclic operators are ergodic and yet have only countably many unimodular eigenvalues. In particular, there exist operators on which are ergodic but admit no Gaussian ergodic measure with full support.
This solves a rather intriguing question, which seems to go back to Flytzanis’ paper [22] and has been very much in the air in the last few years.
As for the “individual” viewpoint, we will elaborate on the kind of operators constructed by the third named author in [39]. Recall that the main result of [39] is the existence of chaotic operators on spaces which are not -frequently hypercyclic. In the present paper, we describe a general scheme allowing to produce, among other things,
operators on which are chaotic and frequently hypercyclic but not ergodic;
- -
operators on which are chaotic and -frequently hypercyclic but not frequently hypercyclic.
These are the first examples of such kinds of operators living on a Hilbert space, even if one dispenses with the chaoticity assumption: no examples of frequently hypercyclic non-ergodic operators, nor of -frequently hypercyclic non frequently hypercyclic operators living on a Hilbert space were known before. As already mentioned, up to now the only available examples were operators acting on .
Moreover, our constructions will also enable us to improve the main result of [39] by showing that there exist
operators on which are chaotic and topologically mixing but not -frequently hypercyclic.
One point is especially worth mentioning regarding these results: it is not at all accidental that all the operators we construct turn out to be chaotic. On the contrary, our proofs rely heavily on new criteria for frequent hypercyclicity and -frequent hypercyclicity (stronger than the usual ones), in which the periodic points play a central role.
In this part of the paper, our constructions and arguments are rather technical, and it is not at all clear that they could be by-passed by suitable Baire category arguments. In fact, as already suggested above, there are reasons to believe that technicalities are unavoidable here: since -frequent hypercyclicity, frequent hypercyclicity and ergodicity are atypical properties, it seems difficult to distinguish them using simply the Baire category theorem; unless one restricts oneself to some cleverly chosen special class of operators, which has yet to be found.
The basic questions we address in this paper can also be considered from a third point of view, which is that of descriptive set theory. Indeed, once it is known (by any argument) that two properties of linear operators are not the same, i.e. that two classes of operators are distinct, it is natural to wonder if a stronger conclusion might hold true, namely that these classes do not have the same complexity in the sense of descriptive set theory. More generally, it can be a quite interesting problem to determine the exact descriptive complexity of a given class of operators. In this paper, we do so for chaotic operators and for topologically mixing operators. We also obtain a partial result for -frequently hypercyclic operators, whose proof relies on our general scheme for constructing operators with special properties.
We finish this overview by pointing out that the difficulty of the existence results presented in this paper is specifically connected with the linear setting, and that the corresponding results are essentially trivial if one moves over to the broader setting of Polish dynamical systems (i.e. dynamical systems of the form , where is a continuous self-map of a Polish space ; see for instance the book [24], as well as [17] for more on linear systems as special cases of Polish systems). Indeed, extending the definitions of frequent and -frequent hypercyclicity to the Polish setting in the obvious way, it is not difficult to see that frequently hypercyclic non-ergodic Polish dynamical systems do exist, as well as -frequently hypercyclic Polish dynamical systems which are not frequently hypercyclic. Here are two simple examples.
First, consider an irrational rotation of the unit circle , and denote by the normalized Lebesgue measure on . Then is an ergodic dynamical system for which all points are frequently hypercyclic. Let be a compact subset of which has empty interior and is such that , and consider the set . Then is a dense subset of which is -invariant, and is thus a frequently hypercyclic Polish dynamical system. But, as (by ergodicity) and is uniquely ergodic, admits no invariant measure at all. This shows in particular the existence of frequently hypercyclic Polish dynamical systems which are not ergodic. This example is due to B. Weiss (private communication).
Now, consider a frequently hypercyclic operator on . Then, the set of all -frequently hypercyclic vectors for is comeager in ([9], see also [14]), whereas the set of frequently hypercyclic vectors for is meager ([40], [9]). Hence is comeager in . Let be a dense subset of contained in , and set . Then is a dense subset of which is -invariant, so that is a Polish dynamical system. Since is still contained in , all points of are -frequently hypercyclic for but none of them is frequently hypercyclic. In particular, the Polish dynamical system is -frequently hypercyclic but not frequently hypercyclic.
1.2. Background and notations
In this subsection, we recall some well-known definitions, referring to [7] or [31] for more details. We also fix some notations that will be used throughout the paper.
1.2.1. Transitivity, mixing and weak mixing
If , we set, for any subsets of ,
[TABLE]
It is well-known that is hypercyclic if and only if it is topologically transitive, i.e. for all open sets . A stronger property is topological mixing: is topologically mixing if all sets are cofinite rather than just non-empty; that is, for any pair of non-empty open sets in , one has for all but finitely many . In between transitivity and mixing is topological weak mixing: an operator is topologically weakly mixing if is hypercyclic on ; in other words, if for all non-empty open sets . We set
[TABLE]
so that
[TABLE]
It is easy to see (for example by considering weighted backward shifts) that the inclusion is proper. As already mentioned, the inclusion is also proper: this was first proved by De La Rosa and Read in [18] for operators living on a suitably manufactured Banach space, and then in [6] for Hilbert space operators. Recall also that, according to a nice result of Bès and Peris [11], the topologically weakly mixing operators are exactly those satisfying the so-called Hypercyclicity Criterion.
1.2.2. Chaos
An operator is said to be chaotic in the sense of Devaney if it is hypercyclic and its periodic points are dense in (in the linear setting, hypercyclicity automatically implies sensitive dependence on the initial conditions). We set
[TABLE]
It is not completely obvious, but nonetheless true, that chaotic operators are topologically weakly mixing (see e.g. [7, Ch. 4]). In other words,
[TABLE]
1.2.3. Ergodic-theoretic properties
In this paper, the word “measure” will always mean “Borel probability measure”. A measure on is invariant for some operator if ; and an invariant measure for is non-trivial if (note that the point mass is invariant for any since ). Also, a measure on is said to have full support if for every non-empty open set . This means exactly that the topological support of is equal to the whole space .
An operator is said to be ergodic if it admits an invariant measure with full support with respect to which it is ergodic, i.e. such that for every Borel subset of satisfying , we have or . If the measure can be taken to be Gaussian, we say that is ergodic in the Gaussian sense, or -ergodic. The operator is said to be mixing if it admits an invariant measure with full support with respect to which it is strongly mixing, i.e. such that for any Borel sets . Mixing in the Gaussian sense is defined as expected. The corresponding notations are the following:
[TABLE]
Since the measures involved are required to have full support, it is obvious by definition that ergodic operators are hypercyclic and that mixing operators are topologically mixing. That is,
[TABLE]
These inclusions are proper: for example, if is any compact weighted backward shift on the Hilbert space , the operator is topologically mixing (see [7] or [31]) but not ergodic: its spectrum is reduced to the point , so that it is by [48] not even -frequently hypercyclic. We also set
[TABLE]
Thus, for example, any operator admitting a non-zero periodic point belongs to , and any chaotic operator lies in . So we have
[TABLE]
This inclusion cannot be reversed: there exist even -ergodic operators on which are not chaotic (see [3] or [7, Section 6.5]).
At this point, we would like to stress that in the present paper, we will mostly focus on invariant measures which are not required to be Gaussian. This is a true change of point of view compared with earlier works on the ergodic theory of linear dynamical systems (see for instance, among other works, [22], [4], [5], [7] …), where Gaussian measures play a central role. But in retrospect, this is in fact quite natural: as we shall see in Section 4, it turns out that within a certain natural class of upper-triangular operators on , the ergodic operators which are not ergodic in the Gaussian sense are typical. Note also that non-Gaussian measures already played an essential role in such works as [41], [25] or [28] for instance.
1.2.4. Frequent hypercyclicity and -frequent hypercyclicity
Let . For any and , set
[TABLE]
The operator is said to be frequently hypercyclic if there exists some such that, for any open set , the set has positive lower density; and is -frequently hypercyclic if there is some such that all sets has positive upper density.
Frequent hypercyclicity was introduced in [4] and rather extensively studied since then (for instance in [9], [13], [27], [29], [39], [48] …). The study of upper-frequent hypercyclicity is more recent. This notion was introduced by Shkarin in [48]. Until the last few years, it has perhaps been unfairly considered as somehow “less interesting”, probably because of a lack of examples or results exhibiting truly different behaviors between frequently and -frequently hypercyclic operators. However, despite a formal similarity in the definitions, there are some important differences between frequent and -frequent hypercyclicity. For example, the set of frequently hypercyclic vectors for a frequently hypercyclic operator on is always meager in ([40], [9]) whereas if is a -frequently hypercyclic operator on , the set of all -frequently hypercyclic vectors for is comeager in ([9]; see also [14] for more along these lines). This may lead to believe that -frequent hypercyclicity is a typical property while frequent hypercyclicity is not; which is in fact a wrong intuition: as we shall see, both properties are atypical. Yet, these properties are not equivalent: Bayart and Rusza exhibited in [9] examples of -frequently hypercyclic operators on which are not frequently hypercyclic. The notion of -frequent hypercyclicity was further studied by Bonilla and Grosse-Erdmann in [14], and it plays an important role in the present paper.
One can go one step further and study -hypercyclic operators, for a given family of subsets of (see in particular [12] and [14]); but we will not follow this quite interesting route here. As for the notations, we set
[TABLE]
Although the definitions of frequent hypercyclicity and -frequent hypercyclicity make no explicit reference to measures, they have a partly “metrical” flavour; and indeed, invariant measures are quite relevant here. First, it follows easily from the pointwise ergodic theorem that ergodic operators are frequently hypercyclic. Moreover, it is shown in [29] that -frequently hypercyclic operators on admit invariant measures with full support (this is in fact true for operators living on any reflexive Banach space). So we have
[TABLE]
The inclusion is proper: as already mentioned, it is proved in [39] that there exist even chaotic operators on which are not -frequently hypercyclic. As we shall see in Section 6 (Theorems 6.22 and 6.29), the other two inclusions are also proper.
1.2.5. Properties related to eigenvalues
If , we denote by the set of all unimodular eigenvectors of , i.e. eigenvectors of whose associated eigenvalue has modulus . We say that has a spanning set of unimodular eigenvectors if ; and that has a perfectly spanning set of unimodular eigenvectors if for any countable set , where denotes the unit circle in , we have . The notations are as follows:
[TABLE]
We also define
[TABLE]
Properties of unimodular eigenvectors of an operator are closely related to those of invariant Gaussian measures for (see for instance [4], [5], [7]): indeed, it turns out that in fact
[TABLE]
so that there was in fact no need for introducing new notations. However, things are not that neat on arbitrary Banach spaces: see [5], [7] or [8]. This is one of the advantages of working on Hilbert spaces, or at least on “nice” Banach spaces.
1.2.6. Distributional chaos
Distributional chaos was first defined by Schweitzer and Smítal in [49], under the name strong chaos, for self-maps of a compact interval, and several definitions have been proposed afterwards in the context of general metric spaces. There is no need to recall these definitions here, because things simplify greatly in the linear setting: as shown in [10], an operator is distributionally chaotic if and only if it admits a distributionally irregular vector, i.e. a vector for which there exist two sets and of integers, both of upper density in , such that
[TABLE]
The corresponding notation is the following one:
[TABLE]
A related class is that of densely distributionally chaotic operators, i.e. of operators which admit a dense set of distributionally irregular vectors. We use the following notation for this set:
[TABLE]
Obviously, .
1.2.7. The parameter
Ergodicity and distributional chaos are closely related to a natural parameter introduced in [29]. This quantity is associated to any hypercyclic operator , and essentially represents the maximal frequency with which the orbit of a hypercyclic vector for can visit a ball in centered at [math]. The precise definition is as follows: for any , we have
[TABLE]
It is shown in [29] that in fact \overline{\rm dens}\,\mathcal{N}_{T}\bigl{(}x,B(0,\alpha)\bigr{)}=c(T) for a comeager set of vectors ; so we have in particular (for any )
[TABLE]
Note also that if the operator is -frequently hypercyclic.
The last class of operators which we introduce is that of operators such that is maximal:
[TABLE]
Since, as proved in [10], the set of all distributionally irregular vectors for a given operator is comeager in , it is clear that . Moreover, it is shown in [29] (more accurately, half in [9] and half in [29]) that ergodic operators are densely distributionally chaotic: . So we have
[TABLE]
It is left as an open question in [29] to determine whether , or whether at least . It follows from the proof of the main result of [39] that the first inclusion does not hold true: the chaotic operator constructed there belongs to (as does any chaotic operator on ) but satisfies . However this operator is not frequently hypercyclic, and so the examples of [39] do not disprove the second inclusion. We will nonetheless show in the present paper that this inclusion does not hold true either (Theorem 6.22).
1.2.8. A last notation
If is a class of operators on , then, for any , we set
[TABLE]
1.3. Organization of the paper
In Section 2, we recall a few basic facts concerning the topologies SOT and , and then prove some “typicality” results in the spaces and , with respect to the Strong∗ Operator Topology . These results can be summarized as follows: for any , an -typical is topologically weakly mixing but not mixing (Propositions 2.16 and 2.17), has no eigenvalues (Corollary 2.23), admits no non-trivial invariant measure (Theorem 2.29), but is densely distributionally chaotic (Proposition 2.37).
Section 3 is a digression, in which we discuss the descriptive complexity of some of the families of operators introduced above. We show that for any , and are Borel subsets of of class exactly , aka (Propositions 3.2 and 3.4), whereas is Borel of class at most , and neither nor (Corollary 3.12). We also show that some rather natural classes of operators defined by dynamical properties are non-Borel in ; for example, the family of all operators admitting a hypercyclic restriction to an invariant subspace is non-Borel (Proposition 3.23), as well as the family of distributionally chaotic operators (Proposition 3.28). In contrast, the class of densely distributionally chaotic operators is (Proposition 2.37).
In Section 4, we consider ergodicity properties of upper triangular operators, this time with respect to the Strong Operator Topology SOT. We show first that for any , an SOT-typical upper triangular operator is ergodic in the Gaussian sense (Proposition 4.4). On the other hand, we essentially show in Theorem 4.9 that an SOT-typical operator of the form “diagonal + backward shift” is ergodic but admits only countably many eigenvalues (and hence is ergodic but not ergodic in the Gaussian sense).
In Section 5, we prove several criteria for an operator to be -frequently hypercyclic or frequently hypercyclic. These criteria are rather different in spirit from the by now classical Frequent Hypercyclicity Criterion ([13]) or from the more recent criteria which can be found in [12] or [14], since they rely explicitly on the existence of many periodic (or almost periodic) vectors for the operator . However, our criterion for frequent hypercyclicity turns out to be stronger than the classical one: indeed, any operator satisfying the so-called Operator Specification Property also satisfies the assumptions of our criterion, whereas it is known [2] that operators satisfying the assumptions of the classical Frequent Hypercyclicity Criterion have the Operator Specification Property (but not conversely, see [2]). In the present paper, these criteria for -frequent hypercyclicity and frequent hypercyclicity are instrumental: we use them in order to simplify the proofs of several later results (more precisely, Corollary 5.11, Theorem 5.14 and Theorem 5.31 will be used in the proofs of the main results of Section 6). However, we believe that these criteria might be useful in other situations as well; and for this reason we have stated them in the setting of general Banach spaces.
In Section 6, which is by far the the most technical part of the paper, we develop a general machinery for producing hypercyclic operators with special properties. This machinery is very much inspired from the construction of [39], but things are done here in greater generality. Again, we hope that this approach could be useful to solve other questions as well. The operators we construct depend on a number of parameters, and we are able to determine in a rather precise way how the parameters influence on -frequent or frequent hypercyclicity, ergodicity or topological mixing. This allows us to produce the examples we are looking for. The main results we obtain are the existence of chaotic and frequently hypercyclic operators which are not in and hence are not ergodic (Theorem 6.22), the existence of chaotic and -frequently hypercyclic operators which are not frequently hypercyclic (Theorem 6.29), and the existence of chaotic and topologically mixing operators which are not -frequently hypercyclic (Theorem 6.34).
We conclude the paper (Section 7) with a short list of possibly interesting questions.
Acknowledgements: We are grateful to Alfred Peris for interesting discussions, and also for kindly allowing us to reproduce here his proof of Theorem 5.36.
2. Typical properties of hypercyclic operators
2.1. The strong and strong topologies
The Strong Operator Topology (SOT) on is defined as follows: any has a neighborhood basis consisting of sets of the form
[TABLE]
where and . Thus, a net tends to with respect to SOT if and only if in norm for every .
The second (and perhaps a little less well-known) topology we will use is the Strong Operator Topology (), which is the “self-adjoint” version of SOT. A basis of -neighborhoods of is provided by the sets of the form
[TABLE]
where and . In other words, a net tends to with respect to if and only if and . Obviously, SOT is coarser than , which is in turn coarser than the norm topology. The topologies induced by SOT and on any closed ball are easily seen to be Polish, i.e. separable and completely metrizable (see e.g. [43, Section 4.6.2]). This will be of primary importance for us.
The following simple fact will be used repeatedly, sometimes without explicit mention.
Fact 2.1**.**
Let . If is endowed with either SOT or , then the map is continuous from into . Consequently, for any fixed integer , the map is -continuous and -continuous from into .
Démonstration.
It is enough to check the statement involving the SOT topology; so we have to show that for any fixed vector , the map is continuous from into . Since the map is continuous from into , it suffices to check that the map is continuous from into . Now, for any operators and any vectors , we have
[TABLE]
The result follows immediately. ∎
It is easy to deduce from this fact that is SOT- in for any . Indeed, it follows from Fact 2.1 that if and are two non-empty subsets of with open, then, for any fixed , the set
[TABLE]
is SOT-open in . Now, fix a countable basis of non-empty open sets of , and observe that
[TABLE]
This proves that is in with respect to SOT. Since the topology is finer than the topology SOT, is also in with respect to . Thus, we may state:
Fact 2.2**.**
For any , is SOT- and - in . Hence and are Polish spaces.
We now state a less immediate fact, which will be proved in Corollary 2.12 below. The corresponding (weaker) SOT statement can be found in [16], and its analogue undoubtedly would have been proved there if there had been any need to do so.
Fact 2.3**.**
For any , is dense in . Hence, is comeager in , both for the SOT and the topology.
Note that this result is indeed not completely trivial. For example, since the map is a homeomorphism of , it immediately implies that a typical operator is hypercyclic and has a hypercyclic adjoint. In particular, this argument proves that there exist hypercyclic operators whose adjoint is also hypercyclic; which is a classical result of Salas obtained via an explicit construction in [44].
We deduce immediately from Fact 2.3 that the word “typical” has the same meaning in the whole of or in the subclass of hypercyclic operators in . This will be used repeatedly below, sometimes without explicit mention.
Fact 2.4**.**
Let be a class of operators on , and let . If is a dense subset of , where is either SOT or , then is a dense subset of ). Conversely, if and if is dense (resp. comeager) in , then is dense (resp. comeager) in .
Démonstration.
This is obvious: if is dense in then, by Fact 2.2, Fact 2.3 and the Baire category theorem, is dense in , and hence in ; and likewise for the converse. The “comeager” case follows from the “” case. ∎
Here is a last fact concerning the topology that will be quite useful for us. Note that the corresponding statement for SOT is false. This is an important difference between the two topologies, which explains in particular why we will encounter some subsets of which are with respect to and not with respect to SOT.
Fact 2.5**.**
Let us denote by the weak topology of . If is a bounded subset of , then the map is continuous from into .
Démonstration.
We have to show that for any fixed vector , the map is continuous on . The key point is that one can separate from by writing (this trick would be useless, of course, if we were working with the SOT topology).
Let be a net in converging to some element ; that is, and . Then
[TABLE]
Since the net is bounded in norm and , this shows that . ∎
2.1.1. Why SOT and ?
There are other natural topologies on , most notably the operator norm topology, of course, and the Weak Operator Topology (WOT). The norm topology is not very well-suited for Baire category arguments, mainly because it is much too strong; in particular, the lack of separability seems unacceptable. The topology WOT is better behaved in this respect, being Polish on any closed ball . However, since we are interested in typical properties of hypercyclic operators, it seems better to consider topologies with respect to which is comeager in for any . This is definitely not the case for WOT. Indeed, it is proved in [19] that a typical element of is unitary. It follows that a WOT-typical is a multiple of a unitary operator and hence not hypercyclic. Incidentally, the operator norm topology has the same “drawback”, in an even stronger way: the hypercyclic operators (actually, even the cyclic operators) are in fact nowhere dense in ; see [7, Section 2.5].
So we will consider neither the operator norm topology nor the Weak Operator Topology in this paper. Actually, when working in the whole of or , , we will always use rather than SOT. Indeed, with respect to SOT, a result from [21] gives a rather complete picture as far as typical properties are concerned: a typical element of is unitarily equivalent to the operator , the countable direct -sum of the unilateral backward shift on . It follows that for every , the class of operators which are unitarily equivalent to is comeager in . Now, the dynamical properties of the operators , are quite strong and very well understood: these operators are mixing in the Gaussian sense (and hence ergodic and topologically mixing), densely distributionally chaotic, and they have nearly any other strong dynamical property one might think of. This explains why, when trying to determine which properties are typical within the class of all hypercyclic operators, we will use rather than SOT. The situation in this setting is more involved, and thus leads to more interesting results.
On the other hand, we will see that within specific subclasses of consisting of upper triangular operators, the topology SOT becomes much more useful. This is not really surprising, since triangularity is not exactly a self-adjoint property.
2.2. How to prove density results
For future reference, we state here a simple criterion for a class of operators to be dense in or , . We will use it repeatedly in the rest of the paper.
Lemma 2.6**.**
Let be a class of operators on , and let . Let also be an orthonormal basis of , and for each , denote by the finite-dimensional subspace of . Suppose that the following property holds true:
for every , every operator with and every , there exists an operator such that
[TABLE]
Then is dense in . If ( ‣ 2.6) is replaced by its self-adjoint version
[TABLE]
then is dense in .
Note that there is a slight abuse of notation in the statement of Lemma 2.6: we consider the operator as an operator on by identifying it with , where is the orthogonal projection of onto .
Démonstration.
We will prove the assertion concerning the -topology, the proof of the SOT statement being exactly the same. Fix , , and . Without loss of generality, we can suppose that . We are looking for an operator such that
[TABLE]
Since for every , and since every vector can be approximated by a finite linear combination of the basis vectors , there exists an integer sufficiently large such that (1) above holds true as soon as
[TABLE]
For each , consider the operator Observe that . By our assumption, there exists such that
[TABLE]
Now
[TABLE]
and
[TABLE]
for every . Since for the SOT topology, one can choose so large that
[TABLE]
from which the conclusion of Lemma 2.6 follows. ∎
Remark 2.7**.**
It is sometimes more convenient to endow with an orthonormal basis indexed by rather than by . In this case, the corresponding version of Lemma 2.6 reads as follows.
For each , denote by the finite-dimensional subspace of . Suppose that the following property holds true:
for every , every operator with and every , there exists an operator such that
[TABLE]
Then is dense in . If ( ‣ 2.6) is replaced by its self-adjoint version
[TABLE]
then is dense in .
2.3. Construction of mixing operators, and density of
In order to show that a property is typical, we first need to prove the density of the set of operators satisfying it. In this subsection, we show that the class is dense in for any . This will be achieved by considering some perturbations of weighted unilateral or bilateral weighted shifts with respect to some orthonormal basis of . We show that these operators admit spanning eigenvector fields which are analytic in a neighborhood of the unit circle, and hence are mixing in the Gaussian sense. The precise statement we will use is the following.
Fact 2.8**.**
Let . Assume that there exists a connected open set with and a family of holomorphic maps, , such that for every and every , and . Then is mixing in the Gaussian sense.
Démonstration.
Recall that we denote by the set of all unimodular eigenvectors of , and by the eigenvalue associated to . By [4, Th. 3.29], it is enough to show that for any Borel set of Lebesgue measure [math], we have . Let be orthogonal to the set . Then for every and every . Since the functions are holomorphic on , and since certainly has an accumulation point in (because has Lebesgue measure [math]), it follows that is orthogonal to all vectors , , , and hence that . This concludes the proof. ∎
Remark 2.9**.**
The assumptions of Fact 2.8 imply that the operator is also chaotic: since the roots of unity contained in have an accumulation point in , this follows as above from the identity principle for holomorphic functions.
Let us first consider perturbations of unilateral weighted shifts. Let be an orthonormal basis of the Hilbert space , and let be a unilateral weight sequence, i.e. a bounded sequence of positive real numbers. Let be an integer, and let be an operator acting on the finite-dimensional space . We define a bounded operator on by setting
[TABLE]
Proposition 2.10**.**
Let be a unilateral weight sequence, , and . Suppose that for every ,
[TABLE]
Then the operator is mixing in the Gaussian sense. Besides, is also chaotic.
Démonstration.
Solving formally the equation , where and , one gets the following identities:
[TABLE]
From this (setting ) we infer that the eigenvectors of associated to the eigenvalue must be given by the formula
[TABLE]
where is a non-zero vector of . Conversely, if belongs to is such that the above formula makes sense, then is an eigenvector of with associated eigenvalue . It follows that the complex number is an eigenvalue of as soon as
[TABLE]
which holds true whenever . In this case, the eigenvector field is well-defined and holomorphic on the open disk for every . Note that our assumption implies that , so that the disk contains .
By Fact 2.8 and Remark 2.9, in order to show that belongs to and is chaotic, it suffices to check that the eigenvectors , , , span a dense subspace of . So let be such that for every and every . Writing as , this means that
[TABLE]
for every . It follows that each vector
[TABLE]
is orthogonal to , i.e. belongs to the closed linear span of the vectors , . In other words,
[TABLE]
Consider now the open subset of , where denotes the spectrum of . Applying the operator to the previous equation, we obtain that for every :
[TABLE]
Since the expression on the right hand side of (3) defines an antiholomorphic map on and since the disk contains (recall that by assumption), it follows that the map extends antiholomorphically to the whole complex plane. But as , so the function must vanish identically on by Liouville’s Theorem, which is possible only if .
Going back to (3), the fact that yields that for every and every , and that for every . Thus , and this concludes the proof of Proposition 2.10. ∎
Remark 2.11**.**
The definition of the operator shows that every eigenvalue of is also an eigenvalue of . This explains why the conditions for every are not sufficient to ensure that be mixing in the Gaussian sense. Indeed, if is such that , then is an isolated eigenvalue of with , and this prevents from being hypercyclic. This is to be compared with Remark 2.15 below.
From Proposition 2.10 we easily deduce a basic density result, which gives in particular the promised proof of Fact 2.3 above.
Corollary 2.12**.**
For every , the class is dense in .
Démonstration.
We are going to apply Lemma 2.6. So, let be an orthonormal basis of , and let us fix , an operator with and . Let also be a small positive number to be specified below. We define a weight sequence as follows:
[TABLE]
and we consider the associated operator acting on . Identifying with , we have for every :
[TABLE]
so that . It follows that if , then
[TABLE]
The assumption of Proposition 2.10 is clearly satisfied, so that belongs to the class .
To estimate the norm of , note that for every , we have
[TABLE]
so that
[TABLE]
Since , it follows that if is sufficiently small. So belongs to , and Lemma 2.6 now allows us to conclude the proof of Corollary 2.12. ∎
Remark 2.13**.**
Corollary 2.12 does not state that is comeager in . Indeed, we will prove below that and are actually meager in .
Let us now turn to a bilateral analogue of Proposition 2.10, which we state as Proposition 2.14 below. Let be an orthonormal basis of the space , and let be a bilateral weight sequence, i.e. a bounded sequence of positive real numbers indexed by . For any integer , we write , and let be a bounded operator on . We define a bounded operator on by setting
[TABLE]
Proposition 2.14**.**
Let be a bilateral weight sequence, and . Suppose that for every ,
[TABLE]
and
[TABLE]
Then is mixing in the Gaussian sense and chaotic.
Démonstration.
The proof is so similar to that of Proposition 2.10 that we will not give it in detail. A complex number is an eigenvalue of as soon as the series
[TABLE]
are convergent for all . If we define and , our assumption implies that . Any complex number belonging to the annulus is an eigenvalue of , and the eigenvectors of associated to have the form
[TABLE]
where . Since , the annulus contains , and an argument similar to the one given in the proof of Proposition 2.10 show that belongs to . Indeed, if is such that for every and every , then
[TABLE]
for every and every , from which it follows that
[TABLE]
for every and every . Hence , and Proposition 2.14 is proved. ∎
Remark 2.15**.**
The description of the eigenvectors of given in (2.3) above shows that if the two series
[TABLE]
are divergent for every , then the operator has no eigenvalue, whatever the choice of . This observation will be useful for the proof of Proposition 2.22 below. It is also interesting to observe that the assumptions on and in Proposition 2.14 do not involve the operator , contrary to what happens in Proposition 2.10. This is coherent with the fact that the eigenvalues of do not necessarily appear as eigenvalues of , while they do appear as eigenvalues of .
2.4. Topological weak mixing and topological mixing
In this subsection, we show that topological weak mixing is a typical property, whereas topological mixing is atypical. In view of the corresponding well-known analogues in ergodic theory due to Halmos and Rohlin (see e.g. [33, pp. 77-80], and [20] for a more general result), this should not be surprising at all.
Proposition 2.16**.**
For every , the class is a dense (and hence comeager) subset of .
Démonstration.
That is follows from the fact that is in , and the -continuity of the map . Since operators in are topologically mixing, density follows from Corollary 2.12. ∎
Proposition 2.17**.**
For every , the class is meager in the space .
The proof of Proposition 2.17 relies on Lemmas 2.30 and 2.31, which belong to the forthcoming Section 2.6. We nonetheless prefer to present things in this order, because the purposes of Lemmas 2.30 and 2.31 will appear more clearly in our proof of the typicality of operators without non-trivial invariant measures.
Démonstration.
By Fact 2.4, it is enough to show that the class is meager in . Let be a non-trivial closed ball in . We certainly have
[TABLE]
where
[TABLE]
Each set is closed in . Indeed, we may write
[TABLE]
Since is weakly closed in , the condition “” defines a closed subset of by Facts 2.1 and 2.5; and since is weakly compact, this shows that is closed in . To conclude the proof, it is enough to show that for some suitable choice of the ball , the closed sets have empty interior in ; or, equivalently, that the open sets
[TABLE]
are dense in . We choose for the ball , where satisfies . Then contains the set of Lemma 2.30 below, so it is dense in by Lemma 2.31. ∎
Remark 2.18**.**
The same proofs would show that topological weak mixing is typical and topological mixing is atypical for operators on spaces. It would be interesting to know if this is still true on every Banach space with separable dual. In this respect, it is worth mentioning that there exist on any separable Banach space hypercyclic operators which are not topologically mixing ([30]).
2.4.1. Some illustrations
In this subsection, we present some consequences of Propositions 2.16 and 2.17.
First, we have the following amusing fact: a typical operator , , satisfies the Hypercyclicity Criterion but not Kitai’s criterion.
In the same spirit, it follows from Proposition 2.16 and the Baire Category Theorem that a typical is such that and are both topologically weakly mixing; but no operator with this property can be topologically mixing since otherwise would be hypercyclic on , which can never happen.
Here is now a less immediate consequence of the comeagerness of , which is a partial strengthening of the main theorem of [53]. This result could also be easily deduced from [26, Th. 4.1], the proof of which is, however, quite different.
Proposition 2.19**.**
Let be a linear subspace of with countable algebraic dimension. For any , the set of all such that every vector is hypercyclic for is comeager in .
Démonstration.
Let us first recall that any topologically weakly mixing operator on is in fact totally hypercyclic, which means that for any , the -fold product operator is hypercyclic on . This is a classical result, which has nothing to do with linearity; see e.g [24], or [7, Th. 4.6].
For any finite sequence of vectors of , we will denote by the associated Gram matrix,
[TABLE]
and we define \mathcal{H}_{\bf f}:=\bigl{\{}\mathbf{x}=(x_{1},\dots,x_{N})\in\mathcal{H}^{N};\;{\rm Gram}(\mathbf{x})={\rm Gram}(\mathbf{f})\bigr{\}}. Note that is a closed subset of , and hence a Polish space. The key point in the proof of Proposition 2.19 is the following fact.
Fact 2.20**.**
Let be a finite sequence of linearly independent vectors in . For any operator , the set is dense in .
Proof of Fact 2.20.
Let be arbitrary. We are looking for some -tuple such that lies in and is very close to . Since is dense in , one can first choose very close to .
Note that the -tuple consists of linearly independent vectors since the Gram matrix is invertible; and is linearly independent as well since belongs to . Let us denote by and the sequences obtained by applying the Gram-Schmidt orthonormalization process to and respectively. Then and are very close to each other, provided and are sufficiently close. Now, define as follows:
[TABLE]
We have by definition
[TABLE]
so that belongs to . Moreover, each vector , , is very close to , so that is very close to . It remains to show that belongs to .
Since the vectors are linearly independent (because belongs to ) and belong to , they form a basis of . So we may write each vector as
[TABLE]
where the matrix is invertible. Now, let be non-empty open sets in , and define
[TABLE]
Since the matrix is invertible, this is a non-empty open subset of . As , one can find an integer such that , which means that by the definition of . This shows that belongs to . ∎
From Fact 2.20, we now deduce:
Fact 2.21**.**
Let . For any finite family of linearly independent vectors in , the set of all such that belongs to is comeager in .
Proof of Fact 2.21.
Let us consider the set
[TABLE]
This is a subset of . Moreover, Fact 2.20 asserts that for any , the -section of is dense in , and hence comeager in since this is a set. Since is comeager in , it follows, by the Kuratowski-Ulam Theorem (see for instance [37, Section 8.K]), that there exists at least one (in fact, comeager many) such that the set
[TABLE]
is comeager in .
Now, since belongs to , i.e. for every , one can find a unitary operator such that for every . Then the map maps bijectively onto itself because is unitary, and is a homeomorphism with respect to the topology . Therefore, the set is comeager in . Since by definition belongs to if belongs to , this concludes the proof of Fact 2.21. ∎
We are now ready to conclude the proof of Proposition 2.19. Let be an algebraic basis of . By Fact 2.21, the set
[TABLE]
is comeager in . So it is enough to show that that if belongs to , every non-zero vector is a hypercyclic vector for . Let be a non-empty open set in . Write as , and consider the set
[TABLE]
Since , this is a non-empty open set in . As belongs to , it follows that there exists such that belongs to , which means exactly that belongs to . Hence is indeed a hypercyclic vector for . ∎
2.5. Hypercyclic operators without eigenvalues
The following result shows that operators without eigenvalues are typical.
Proposition 2.22**.**
For any , the class is a dense subset of .
From this and Fact 2.4, we obtain
Corollary 2.23**.**
For any , a typical operator T\in\emph{(\textrm{HC}_{M}(\mathcal{H}),\texttt{SOT}^{})} has no eigenvalues.*
As a matter of fact, Proposition 2.22 is already proved in [21], where typical properties of contraction operators are studied for various topologies (see also [19]). However, since the proof is not that complicated, and in order to keep the paper as self-contained as possible, we outline it below.
Proof of Proposition 2.22.
We divide the proof into two steps. In what follows, we fix .
Fact 2.24**.**
The set is a subset of .
Proof of Fact 2.24.
To any closed ball , we associate the following subset of :
[TABLE]
Let us show that this set is in . To this aim, we endow with the weak topology, and introduce the set
[TABLE]
Then is the projection of on the first coordinate. Moreover, the set is closed in by Fact 2.5. Since the space is (because is compact), it follows that is .
Let now be a sequence of closed balls of not containing the point [math], such that . Then is an set, so that is a set in . ∎
Fact 2.25**.**
The set is dense in .
Proof of Fact 2.25.
Let be an orthonormal basis of . For each , we set and we denote by the orthogonal projection of onto . By Lemma 2.6 and Remark 2.7, it suffices to show that for every , every , with , and every , there exists an operator such that
[TABLE]
Let be a bilateral weight sequence with for every and for every index with . As in the proof of Corollary 2.12, one easily checks that if is sufficiently small, the bilateral weighted shift operator (defined with respect to the basis ) satisfies
[TABLE]
Moreover, if the weight sequence is chosen in such a way that the series
[TABLE]
are divergent for every , then has no eigenvalue by Remark 2.15. So satisfies the required assumptions for a suitable choice of the weight . ∎
The two facts above complete the proof of Proposition 2.22. ∎
Remark 2.26**.**
The fact that we are using the topology is crucial in the proof of Fact 2.24 in order to obtain that the sets above are closed in . The situation turns out to be completely different if one considers the topology SOT instead of . Indeed, it is proved in [21] that an SOT-typical has the property that every with is an eigenvalue of . More precisely, a typical is unitarily equivalent to the infinite-dimensional backward unilateral shift operator. So a typical has the whole disk within the set of its eigenvalues.
Remark 2.27**.**
The proof of Corollary 2.23 is a good example of the usefulness of Fact 2.4 for simplifying arguments of this kind: although it is easy to construct the weight sequence above in such a way that the operator is -close to and has no eigenvalue, it is technically much less obvious to ensure that is additionally hypercyclic. In other words, it would be less easy to prove Corollary 2.23 directly, without using Fact 2.4.
One of the main consequences of Corollary 2.23 is the following result concerning chaotic operators.
Corollary 2.28**.**
For every , is meager in . In other words, a typical hypercyclic operator on is not chaotic.
This is indeed obvious since chaotic operators have plenty of eigenvalues. Note that we are using here the fact that is a complex Hilbert space, so that periodic points are linear combinations of eigenvectors whose associated eigenvalues are roots of unity. Nonetheless, Corollary 2.28 holds true on real Hilbert spaces as well; see Remark 2.34 below.
2.6. Hypercyclic operators without invariant measures
It follows easily from Proposition 2.22 that for any , the operators in admitting a non-trivial invariant measure with a second-order moment form a meager class in . Indeed, if admits an invariant measure such that , then the Gaussian measure whose covariance operator is given by the formula
[TABLE]
is -invariant. Its support is the closed linear span of the support of , and hence is non-trivial. This closed subspace is spanned by unimodular eigenvectors of (see [4] or [7] for details), from which it follows that does not belong to .
The main result of this subsection is that the -typical operator actually admits no non-trivial invariant measure at all.
Theorem 2.29**.**
For every , the set is comeager in the space .
The proof of Theorem 2.29 relies on the next two lemmas.
Lemma 2.30**.**
Let be a closed ball of not containing the point [math]. For any integer , the set
[TABLE]
is open in . Consequently, the set
[TABLE]
is in .
Proof of Lemma 2.30.
The second part of the lemma follows immediately from the first, since . To derive the first part, it is enough to show that if we fix , then the set
[TABLE]
is -open in . If , we may write
[TABLE]
Since the map is continuous on for any by Facts 2.1 and 2.5, the condition “” defines an open subset of . Since is weakly compact, it follows that is indeed -open in , its complement being the projection of a closed subset of along the compact factor . ∎
Lemma 2.31**.**
Let with , and let . Denote by the closed ball . Let also . For any , the open set is dense in .
Proof of Lemma 2.31.
Let us fix an orthonormal basis of with . Our aim being to apply Lemma 2.6, we fix , an operator such that , and . We are looking for an operator such that
[TABLE]
We will define a sequence of operators, with for every , and show that if is sufficiently large, the operator belongs to and satisfies the above estimates. Here denotes as usual the canonical projection of onto .
Let and , to be fixed later on in the proof. For each , consider the operator defined in the following way:
[TABLE]
Thus, in matrix form,
[TABLE]
We note that if and if is sufficiently small, then . Moreover, if is sufficiently small, then
[TABLE]
whatever the choice of the integer . The key of the proof lies in the following simple computation.
Fact 2.32**.**
For every , every such that , and every ,
[TABLE]
Proof of Fact 2.32.
Clearly ; so we may write as
[TABLE]
Hence
[TABLE]
Now belongs to the closed linear span of the vectors , , , for every with . Indeed,
if , a straightforward induction shows that ; 2. -
if , then ; 3. -
if , we have .
Thus , which is the claim of Fact 2.32. ∎
From Fact 2.32, it is not hard to deduce
Fact 2.33**.**
Let be such that . If , then .
Proof of Fact 2.33.
If , then by Fact 2.32
[TABLE]
Moreover, since and belong to , we have and . Hence
[TABLE]
Thus is empty as soon as , which proves our claim. ∎
We now choose the various parameters in this construction in the following order: first we choose such that . Then we choose so small that (5) holds true for every . Lastly, we choose so large that there exist distinct integers with , such that the gaps between two consecutive integers are so large that for every . The operator then satisfies for every . So the operator is such that for every , that is, belongs to ; and by (5) we also have and for every ∎
Proof of Theorem 2.29.
Combining Lemmas 2.30 and 2.31, we obtain that is a dense subset of for every ball , where and . By Fact 2.4, it follows that is a dense subset of for each such ball . Let be a countable family of such balls with the property that
[TABLE]
Then \mathfrak{G}:=\bigl{(}\bigcap_{q\geq 1}\mathfrak{G}_{B_{q}}\bigr{)}\cap\textrm{HC}_{M}(\mathcal{H}) is a dense subset of .
Every element of enjoys the property that for any and any , there exist distinct iterates of which are pairwise distinct. It follows that if is any invariant (probability) measure for , then for every and hence .
Suppose now that admits an invariant measure . Then one can find a closed ball not containing [math] such that . Consider for any the measure on defined by for any Borel subset of . Each such measure is an invariant probability measure on . Moreover, if is sufficiently large then , so that , which is a contradiction. We have thus proved that any operator admits no invariant measure except , and hence that is comeager in . ∎
Remark 2.34**.**
The above proof does not use the fact that is a complex Hilbert space; so Theorem 2.29 holds true as well for real Hilbert spaces. This shows in particular that in the real setting also, the chaotic operators form a meager subset of ; more precisely, a typical operator has no periodic point except [math]. Indeed, any periodic point for an operator gives rise to a “canonical” invariant measure supported on the orbit of , namely , where is a period of .
Remark 2.35**.**
Theorem 2.29 implies a weak form of Proposition 2.22, namely that operators without any eigenvalue of modulus are typical. Indeed, if admits a unimodular eigenvalue and if is an associated eigenvector, then there is a canonical invariant measure supported on , namely the image of the Lebesgue measure on under the map .
Regarding the eigenvalues, one may also note that if is any non-trivial invariant measure for an operator , then m\bigl{(}\ker(T-\lambda)\bigr{)}=0 for every complex number such that . Indeed, if is a measure such that m\bigl{(}\ker(T-\lambda)\bigr{)}>0, one can find a ball not containing [math] such that , where . Since , it is easily checked that has infinitely many pairwise disjoint iterates under , which is not possible since the measure is -invariant.
Since -frequently hypercyclic operators on a Hilbert space always admit an invariant measure with full support by [29], Theorem 2.29 immediately implies:
Corollary 2.36**.**
For every , is meager in . In other words, a typical hypercyclic operator on is not -frequently hypercyclic.
2.7. Densely distributionally chaotic operators
In this subsection, our aim is to show that, for any , the class of densely distributionally chaotic operators in is a dense subset of , from which it follows that the class of distributionally chaotic operators in is a comeager subset of .
Proposition 2.37**.**
For any , the set of densely distributionally chaotic operators in is a dense subset of .
Démonstration.
Recall that an operator is densely distributionally chaotic if and only if it admits a dense set of distributionally irregular vectors, i.e. of vectors for which there exist two sets of integers with upper density such that as along and as along . Since the set of distributionally irregular vectors for can be written as
[TABLE]
where
[TABLE]
it follows that is a dense subset of whenever is densely distributionally chaotic. Denoting by a countable basis of non-empty open subsets of , we infer from this observation that an operator belongs to if and only if
[TABLE]
Using this, we can now prove
Fact 2.38**.**
The set is a subset of .
Démonstration.
It suffices to show that for every , , , and , the set
[TABLE]
is SOT-open. So let belong to this set, and let be an integer such that there exist indices such that for every . If is sufficiently close to for the SOT-topology, we still have for every . Hence . In the same way, the set of operators such that is SOT-open in . This proves Fact 2.38. ∎
The last step of the proof of Proposition 2.37 is the following fact.
Fact 2.39**.**
The set is dense in .
Démonstration.
By a result of [29], every ergodic operator on is densely distributionally chaotic. Since is dense in by Corollary 2.12, it immediately follows that is dense in . ∎
By Facts 2.38 and 2.39, the proof of Proposition 2.37 is now complete. ∎
Since , and since any operator satisfies , Proposition 2.37 has the following immediate consequences:
Corollary 2.40**.**
For any , the set of distributionally chaotic operators in is comeager in .
Corollary 2.41**.**
For any , the set of operators with is comeager in .
There is an alternative approach for proving the comeagerness of , which relies on the next proposition, combined with the Kuratowski-Ulam Theorem.
Proposition 2.42**.**
Let . For every vector , the set
[TABLE]
is comeager in .
By the Kuratowski-Ulam Theorem, Proposition 2.42 implies that the set
[TABLE]
is comeager in ; and this, in turn, implies that the set
[TABLE]
is comeager in . Here, “” means “for quasi-all in the Baire category sense”. This shows that is comeager in .
For the proof of Proposition 2.42, we will need the following lemma.
Lemma 2.43**.**
Let be an ergodic operator on , and let be a -invariant ergodic measure with full support for . For every , , one can find two other ergodic measures and for , both with full support, such that and respectively.
Démonstration.
The proof relies on a dilation argument already used several times in [29]. This argument has been essentially given at the end of the proof of Theorem 2.29, but we repeat it anyway. For any , let be the measure on defined by for any Borel subset of . We have . All these measures are ergodic for and have full support. Moreover, as and as (that is necessarily equal to [math] follows from the ergodicity of with respect to and the fact that ). So there exist , such that and satisfy and . ∎
Proof of Proposition 2.42.
Let us consider the following two subsets of :
[TABLE]
It is not difficult to see that
[TABLE]
Indeed, if , then one can find a set with such that as along ; whereas if , one can find a set with such that as along . So it is enough to show that and are both comeager in . We will actually concentrate on only, the proof for being completely similar.
For any , , we introduce the set
[TABLE]
Reasoning as in the proof of Fact 2.38, we observe that each set is a subset of . We also have:
Fact 2.44**.**
Each set is dense in .
Proof of Fact 2.44.
We will use the density of in , proved in Corollary 2.12 above.
Let be a non-empty open subset of . By Corollary 2.12, we know that is dense in for every . Since is obviously dense in , it follows that contains an operator which is mixing in the Gaussian sense and satisfies . The operator is in particular ergodic. By Lemma 2.43, admits an ergodic measure with full support such that . Birkhoff’s ergodic theorem then implies that the set
[TABLE]
is dense in .
Let now be a small positive number, to be fixed later on in the proof. Since , the density of in implies the existence of an isomorphism of with the following properties: . Consider now the operator . Since , we have if is sufficiently small. Also, belongs to as soon as is sufficiently small. We thus fix such that these two conditions are satisfied. It now remains to prove that belongs to , which will conclude our proof that is dense in .
For every , we have . Since belongs to , it follows that
[TABLE]
Observe now that if , then (as soon as , of course). It follows that
[TABLE]
for all sufficiently large . Hence belongs to , which concludes the proof of Fact 2.44. ∎
The two facts above imply that all the sets are comeager (in fact, dense ) in . Since
[TABLE]
it follows that is comeager as well. The case of being exactly similar, the proof of Proposition 2.42 is now complete. ∎
2.8. Summary
Let us summarize the results obtained so far: for any , an -typical hypercyclic operator
is topologically weakly mixing but not topologically mixing; 2. -
has empty point spectrum, and hence is not chaotic; 3. -
has no non-trivial invariant measure, hence is not -frequently hypercyclic, and a fortiori not ergodic; 4. -
but is densely distributionally chaotic and hence satisfies .
We shall see in Section 4 below that the picture changes drastically when we consider SOT-typical elements of some natural classes of upper-triangular operators with respect to a given orthonormal basis of .
3. Descriptive set-theoretic issues
In what follows, we fix . We have seen in the previous section that , and are meager in . In this section, we are going to show that these classes of operators are also Borel in with respect to SOT and , and we will discuss their exact descriptive complexity in some details. Moreover, we will show that some natural classes of operators defined by dynamical properties are non-Borel in .
Recall the standard notations for Borel classes: , , , and so on. We refer the reader to [37] for more information on the Borel hierarchy.
3.1. Complexity of the families , , and
The following fact will allow us to concentrate mainly on the topology .
Lemma 3.1**.**
The identity map is Baire ; in other words, any -open subset of is SOT-. Therefore, for every countable ordinal , any - subset of is SOT- and any - set is SOT-.
Démonstration.
Since is second-countable, it is enough to show that any basic -open set is SOT-. Therefore, we just have to check that if and , then the set is SOT-. But this is clear since
[TABLE]
and the condition under brackets is SOT-closed. ∎
3.1.1. Complexity of
The complexity of is given by the following proposition:
Proposition 3.2**.**
The set is a subset of , and a “true” subset of , i.e. a set which is not .
Démonstration.
Let us first show that is with respect to , and hence (by Lemma 3.1) with respect to SOT. Let be a countable family of closed balls of whose interiors form a basis of open sets for . Then an operator is topologically mixing if and only if
[TABLE]
For each fixed data , the condition “” is -closed, by weak compactness of , weak closedness of and continuity of the map from into . This shows that is in .
In order to show that is a true subset of , we assume that and we use weighted backward shifts on . It is well-known that a weighted backward shift on is topologically mixing if and only if the weight sequence satisfies
[TABLE]
Let us denote by the set of all sequences of positive integers such that for all . This is a closed subset of the Polish space , and hence a Polish space as well. We need the following fact.
Fact 3.3**.**
The set is a true set in .
Proof of Fact 3.3.
It is known (see e.g. [37, Section 23.A]) that is a true set in . So we just need to find a continuous map such that . In other words, our goal is to associate to each another sequence in such a way that exactly when and, additionally, for all ; and this needs to be done in a continuous fashion. For any , we define as follows: and, for every ,
[TABLE]
It is obvious that for all and that the map is continuous. Moreover, since , it is straightforward to check that if and only if . ∎
Going back to the proof of Proposition 3.2, we associate to each a weight sequence defined as follows:
[TABLE]
Since , we have for all , so that the weighted shift on satisfies . Moreover, the map is clearly continuous from into , and hence the map is continuous from into . Finally, since for all , the shift is topologically mixing if and only if . We have thus constructed a continuous map such that , which proves that is a true subset of by Fact 3.3. ∎
3.1.2. Complexity of
We now consider the class of chaotic operators on .
Proposition 3.4**.**
The set is a subset of , and a true subset of .
Démonstration.
Let us first show that is with respect to , and hence with respect to SOT. Let be a countable family of closed balls whose interiors form a basis of open sets for . By definition, an operator is chaotic if and only if it is hypercyclic and each ball contains a periodic point for . In other words:
[TABLE]
For each fixed pair , the condition under brackets is -closed by continuity of the map and weak compactness of the ball . Therefore, the second half of the condition on the right hand side of the above display defines a set; and since is , it follows that is in .
The proof that is a true set is a little bit more involved. We will use the so-called Kalisch operators, introduced by Kalisch in [36], which display interesting dynamical properties (see for instance [7, Section 5.5.3]). They are defined as follows. Let be the operator defined for every by
[TABLE]
A simple computation shows that for any , the function is an eigenvector of associated to the eigenvalue , and that . Note also that the map is continuous from into . In particular, if is any compact subset of , the closed subspace of spanned by the functions , is -invariant. The Kalisch operator associated to is the operator induced by on . These operators have the following properties.
The spectrum of is exactly equal to .
The operator is hypercyclic if and only if is a perfect set, in which case is actually ergodic in the Gaussian sense.
Recall that a perfect set is a non-empty compact set without isolated points.
It is easy to deduce from these properties the following characterization of compact subsets of such that is chaotic. We denote by the subset of consisting of all roots of unity.
Fact 3.5**.**
Let be a perfect subset of . Then the operator is chaotic if and only if is dense in .
Proof of Fact 3.5.
Since the map is continuous and is a periodic point of if , it is clear that is dense in if is dense in . Since is assumed to be perfect, is hypercyclic, and hence chaotic. Conversely, assume that is not dense in and choose , where . Then is not an eigenvector of since . Since however is an eigenvector of , this means that does not belong to , i.e. that does not belong to . But since and for every , we have
[TABLE]
So is not dense in and is not chaotic. ∎
Here is now a purely descriptive set-theoretic fact, which is certainly well-known but for which we were unable to locate a reference. Here and afterwards, given a compact metric space , we denote by the space of all non-empty compact subsets of endowed with its usual topology, and by the set of all perfect subsets of . Recall that is compact metrizable, and that is a subset of . So is a Polish space. See [37] or [51] for more details.
Fact 3.6**.**
Let be a perfect set such that is dense in . Then the set \mathcal{W}:=\{\Lambda\in\mathcal{K}_{\rm perf}(E);\;\hbox{\Omega\cap\Lambda\Lambda}\} is a true set in .
Proof of Fact 3.6.
Since is in , it is in fact enough to show that is a true set in . That is is easy to check. In order to show that it is a true set, we proceed as follows.
Let us denote by the Cantor space , and by the set of all “rationals” of , i.e. \mathbf{Q}=\{\alpha=(\alpha(i))_{i\geq 1}\in\mathbf{C};\;\hbox{\alpha(i)0}\}. It is known (see [37, Section 23.A]) that the set
[TABLE]
is a true set in . So it is enough to find a continuous map such that .
We first note that if is a perfect set such that is dense in , then is not in . Indeed, on the one hand is easily seen to be dense in because is perfect and is dense in ; and on the other hand is meager in because it is disjoint from the set , which is dense in because is perfect and is countable. Hence cannot be in by the Baire Category Theorem. By Wadge’s Lemma (see [37, Th. 21.14]) applied to the two Borel sets and , it follows that for any perfect set as above, is “-hard”, i.e. one can find a continuous map such that \varphi^{-1}\bigl{(}\mathcal{W}\cap\mathcal{K}(M)\bigr{)}=\mathbf{Q}.
Now let us choose a sequence of pairwise disjoint perfect subsets of such that is dense in for every and the sets accumulate to some point , which means that every neighborhood of contains all but finitely many of the sets . This is possible because is perfect and is dense in . For each , let be a continuous map such that \varphi_{n}^{-1}\bigl{(}\mathcal{W}\cap\mathcal{K}(M_{n})\bigr{)}=\mathbf{Q}. One can then define a map by setting
[TABLE]
Since the sets accumulate to , it is clear that each is indeed a compact subset of and that the map is continuous. Moreover, it is equally clear that is perfect if and only if all the sets are perfect, and that is dense in if and only if is dense in for every . Hence belongs to if and only if belongs to \varphi^{-1}_{n}\bigl{(}\mathcal{W}\cap\mathcal{K}(M_{n})\bigr{)}=\mathbf{Q} for all . We have thus proved that
[TABLE]
which concludes the proof of Fact 3.6. ∎
We need yet one more fact, which is a simple and certainly well-known consequence of the Michael Selection Theorem. The version of this theorem which we use here runs as follows (see [51, Section III.19]).
Let be a zero-dimensional compact space, and let be a complete metric space. Let be a lower semi-continuous map, where denotes the set of all non-empty closed subsets of . Then admits a continuous selection, i.e. there exists a continuous map such that belongs to for every .
Fact 3.7**.**
Let be a zero-dimensional compact metric space. There exists a sequence of continuous maps from into such that for each , the points , , belong to , are pairwise distinct, and the countable set is dense in .
Proof of Fact 3.7.
Since is zero-dimensional, one can choose a countable basis for the topology of consisting of (non-empty) clopen sets. For each , let us denote by the integers such that (since is perfect, there are infinitely many such integers ). As the are clopen sets, it is not hard to see that the functions , , are locally constant. In particular, the map from into is continuous. Since is zero-dimensional, is zero-dimensional as well, and one can apply the version of Michael’s Selection Theorem quoted above to the map . This yields a continuous map from into such that belongs to for every . Now, any perfect set satisfies ; so there exists an integer such that and . Moreover, if we denote by the smallest such integer with these two properties, the map is locally constant, and hence the map \Lambda\mapsto\Lambda\cap\bigl{(}V_{i_{2}(\Lambda)}\setminus V_{j_{2}(\Lambda)}) is continuous from into . Applying Michael’s Selection Theorem a second time, we obtain a continuous map such that \xi_{2}(\Lambda)\in\Lambda\cap\bigl{(}V_{i_{2}(\Lambda)}\setminus V_{j_{2}(\Lambda)}) for every ; in particular, and . Continuing in this fashion, we obtain the required sequence . ∎
We are now in position to prove that is a true set in . Let us first choose a compact set with the following properties: is perfect and zero-dimensional, is dense in and, moreover, . Such a compact set does exist. Indeed, the definition of the operator shows that the norm of the restriction of to the subspace tends to as . Therefore, if is a non-trivial closed arc sufficiently close to then ; so it is enough to take as any perfect set with empty interior contained in and such that is dense in .
Having fixed in this way, let be a sequence of continuous maps given by Fact 3.7, selecting a dense sequence of pairwise distinct points in each perfect set . Note that if , the points are pairwise distinct and form a dense subset of , so that the functions are linearly independent and span a dense subspace of . Applying the Gram-Schmidt orthonormalization process to the sequence , we obtain an orthonormal basis of which depends continuously on , that is, each map is continuous from into .
Let us now fix an orthonormal basis of . For each , denote by the unitary operator defined by setting for every . Since , one can define a map by setting
[TABLE]
Since and are unitarily isomorphic, is chaotic if and only if is, which holds true exactly when is dense in . Thus, we have
[TABLE]
Fact 3.6 will allow us to conclude the proof, provided that we are able to show that the map is continuous from into . This relies on the following observation. For every , let us denote by the orthogonal projection of onto , and by the canonical embedding of into . In other words, .
Claim 3.8**.**
For any , the map is continuous from into ; and for every , the map is continuous from into .
Proof of Claim 3.8.
Fix . Since , we have
[TABLE]
The maps , , being continuous, it suffices to show that the convergence of the above series is uniform (with respect to ) on compact subsets of in order to derive the continuity of the map . Now, we have for any
[TABLE]
In particular, the maps are continuous. Since the sequence is non-increasing, it converges uniformly to [math] on compact subsets of by Dini’s Theorem. This proves the first part of Claim 3.8. The proof of the second part is exactly similar. ∎
Let us now prove that is continuous from into , which amounts to showing that for any , the maps and are continuous. Since , we have
[TABLE]
By Claim 3.8, the map is continuous from into . By Claim 3.8 again, and since the map takes values in a bounded subset of , the map is continuous. Likewise, the map is continuous. The equality , the continuity of and Fact 3.6 now imply that is a true set in . ∎
3.1.3. Complexity of and
In this subsection, we study the descriptive complexity of the sets and in , for any . Although we have been unable to determine the exact complexity of these sets, we obtain some simple upper bounds and some rather non-trivial lower bounds. Our first result reads as follows.
Proposition 3.9**.**
For any , the set is in , hence also in .
Démonstration.
The proof of Proposition 3.9 relies on the following observation, first made in [9] and then developed in [14].
Fact 3.10**.**
Let be a countable basis of (non-empty) open subsets of , and let be a bounded operator on .
- (1)
Assume that belongs to , and let be a -frequently hypercyclic vector for . For each , denote by the upper density of the set . For any integers , define
[TABLE]
Then
[TABLE]
is a dense subset of which consists entirely of -frequently hypercyclic vectors for . 2. (2)
Conversely, assume that there exists a sequence of positive numbers such that for any , the open set defined by the formula above is dense in . Then is -frequently hypercyclic and admits a dense set of -frequently hypercyclic vectors which satisfy
[TABLE]
Let us denote by the set of all positive rational numbers. It follows immediately from Fact 3.10 that an operator belongs to if and only if
[TABLE]
In order to prove that is in , it is enough to check that each set
[TABLE]
is SOT-open in . This is easy: if belong to this set, one can find distinct integers such that belongs to for every . If is sufficiently close to for the SOT-topology, it still satisfies for every , and thus belongs to our set. Hence is in , and thus in as well. ∎
Our next result deals with the complexity of the class .
Proposition 3.11**.**
The set is a difference of sets in , i.e. it can be written as , where and are both - sets. Besides, is neither -, nor - in . More precisely, there is no - set such that , and no - set such that .
As a straightforward consequence of Proposition 3.11, we obtain:
Corollary 3.12**.**
The set is neither , nor in .
Démonstration.
By Proposition 3.11, is not , and it is not because is and is not. ∎
Another immediate consequence is
Corollary 3.13**.**
The set is a true set in .
Démonstration.
By Propositions 3.2 and 3.4, is ; and by Proposition 3.11, it is not . ∎
Proof of Proposition 3.11.
The proof of Proposition 3.11 relies on the following fact, which characterizes in a rather surprising way the -frequently hypercyclic operators within the class of chaotic operators. (We will come back to this at the end of Section 5).
Fact 3.14**.**
Let be a chaotic operator on . Then, is -frequently hypercyclic if and only if .
Proof of Fact 3.14.
This follows from Theorem 5.23, to be proved below. ∎
We now start the proof of Proposition 3.11, which we divide into three parts.
Part 1. is a difference of - subsets of .
By Fact 3.14, we have , where
[TABLE]
Hence, since is - by Proposition 3.4, it suffices to show that is -. Let be a countable basis of non-empty open subsets of , and observe that, by the definition of , an operator belongs to if and only if
[TABLE]
This condition is easily seen to define an - subset of ; and since is in , it follows that is -.
Part 2. There is no - subset of such that .
In order to prove this, we need some preliminary facts. Let us choose a sequence of infinite-dimensional closed subspaces of such that can be decomposed as the orthogonal direct sum of the spaces , i.e. .
Fact 3.15**.**
Let have the form with respect to the decomposition , where for each .
- (i)
If all operators are chaotic and mixing in the Gaussian sense, then so is .
- (ii)
If is -frequently hypercyclic or chaotic, then so are all operators .
Proof of Fact 3.15.
(i) Assume that all operators are chaotic and mixing in the Gaussian sense, and fix for each a mixing Gaussian measure with full support for . Each second-order moment is finite; and by rescaling (i.e. replacing each measure by a measure defined by setting for every Borel subset of , for a suitably small ), we may assume without loss of generality that the series is convergent.
Let us denote by the infinite product measure on the product space . Then
[TABLE]
from which it follows that is in fact concentrated on the set
[TABLE]
We may therefore define a probability measure on as follows: for any Borel subset of ,
[TABLE]
This measure is obviously Gaussian, and it is -invariant because each measure is -invariant. Indeed, for any bounded Borel function , we have
[TABLE]
The measure has full support. Let indeed be a non-empty open set in . Choose a point , and such that . We may assume that belongs to for some , where is so large that
[TABLE]
where \mathbf{H}^{N}:=\Bigl{\{}(x_{n})_{n\geq N}\in\prod_{n\geq N}\mathcal{H}_{n};\;\sum_{n=1}^{\infty}\|x_{n}\|^{2}<\infty\Bigr{\}} and . Then the set contains
[TABLE]
Moreover, if we set \mathbf{A}^{N}:=\bigl{\{}(x_{n})_{n>N};\;\sum_{n>N}\|x_{n}\|<\varepsilon^{2}\}, then by (6). Also, setting and , the fact that the measures have full support implies that . Hence
Finally, is mixing with respect to because each operator , , is mixing with respect to : this can be shown easily by checking that as for any Borel sets whose definition depends on finitely many coordinates only with respect to the decomposition .
Thus, we have shown that is mixing in the Gaussian sense. Finally, since all operators are chaotic, it is easily checked that the periodic points of are dense in ; hence is chaotic.
(ii) Fix . If we denote by the canonical projection of onto , then . Since is continuous and onto (that it has dense range would suffice for our argument), (ii) follows easily: if is a -frequently hypercyclic vector for , then is a -frequently hypercyclic vector for ; and if is chaotic, then and is a dense subset of consisting of periodic vectors for . ∎
Recall that we denote by the set of all “rationals” of the Cantor space ,
[TABLE]
Fact 3.16**.**
There exists a continuous map from into such that the following holds true:
if , then is chaotic and mixing in the Gaussian sense, and hence chaotic and -frequently hypercyclic;
- -
if , then the spectrum of is reduced to the point , and hence is not chaotic and not -frequently hypercyclic.
Proof of Fact 3.16.
The very last part of the second assertion follows from the fact that the spectrum of a chaotic or -frequently hypercyclic operator has no isolated points ([48]).
We assume that , endowed with its canonical basis. For any unilateral weight sequence , let us denote as usual by the associated weighted shift acting on . If , the operator is chaotic and mixing in the Gaussian sense since it admits a spanning holomorphic eigenvector field defined in a neighborhood of the point (see Fact 2.8). On the other hand, if as , then , so that . So, setting , it is enough to find a continuous map from into such that if while as if . Once this is done, the map will enjoy the two properties stated in Fact 3.16. Now, it is quite easy to define such a map : just set, for every ,
[TABLE]
Fact 3.16 is thus proved. ∎
It is now easy to deduce from Facts 3.15 and 3.16 that no subset of with the property that can be -. Let us fix such a set . Decompose as , where the spaces , , are infinite-dimensional. Let us define a map as follows: for every ,
[TABLE]
where the operators , , are given by Fact 3.16. Then is continuous, and Facts 3.15 and 3.16 imply that
[TABLE]
This concludes the proof since is a true subset of , hence not a set.
Part 3. There is no - set such that .
In order to prove this, we will make use of the machinery developed in Section 6 below. The reader may therefore prefer to skip the proof and return to it after reading Section 6.
In what follows, we assume that , endowed with its canonical basis. Let be such that , and choose an even integer large enough to have
[TABLE]
Now, let us denote by the set of all infinite sequence of integers with such that and for every . This is a closed subset of and hence a Polish space. For any element of , we denote by the operator of C*+*-type on defined as follows: for any
[TABLE]
Note that in the terminology of Section 6, looks exactly like an operator of C+,1-type * with , except that in the definition of the weights and the constant has been replaced by . (See Subsection 6.2 for the definition of C+-type operators, and Subsection 6.5 for the definition of C+,1*-type operators.) Observe that if belongs to , then for all . By (7), it follows that for every . Moreover, it is clear that the map is continuous from into .
The key point of the proof that is not - is the following fact, which is actually nothing but a reformulation of Theorem 6.18 in a slightly different setting.
Fact 3.17**.**
Let belong to . Then the operator is always chaotic, and is -frequently hypercyclic if and only if does not tend to [math] as ; in which case is in fact frequently hypercyclic.
Proof of Fact 3.17.
We apply a modified version of Theorem 6.18, with . Accordingly, we set for every
[TABLE]
Since , we have , from which it follows easily that the sequence is non-increasing. Moreover, we have for all . Hence condition (7) above yields that
[TABLE]
The analogues of the “additional assumptions” in Theorem 6.18 are thus satisfied, and it follows that is -frequently hypercyclic on if and only if does not tend to [math] as . ∎
Now, let have the property that . Consider the map defined by setting for every . As already mentioned, the map is continuous from into . By Fact 3.17 we have
[TABLE]
Now, a proof quite similar to that of Fact 3.3 shows that is a true set in , so that is a true set in . Hence cannot be in . ∎
Remark 3.18**.**
Propositions 3.4 and 3.11 together formally yield the statement that, since is not while is in , the class is not void, i.e. there exist chaotic operators in which are not -frequently hypercyclic. However, there is nothing magic here: the proof of Proposition 3.11 relies heavily on a construction, carried out in Section 6, of explicit chaotic operators which are not -frequently hypercyclic.
Remark 3.19**.**
The proof of Proposition 3.11 has established that the set is a true - subset of .
Remark 3.20**.**
An “alternative” proof of the Borelness of could run as follows. According to Fact 3.10, an operator belongs to if and only if quasi-all vectors (in the Baire category sense) are -frequently hypercyclic for . So, for , we may write
[TABLE]
Since the relation B(T,x)\longleftrightarrow\hbox{(x,,\mathcal{U}T) } is Borel in the product space , and since the category quantifier preserves Borelness (see for instance [37, Section 16.A]), it follows that is Borel in .
Remark 3.21**.**
It is likely that and are both true subsets of , i.e. but not , that is a true set in , and that is a true difference of sets in . However, we have been unable to prove any of these facts.
Remark 3.22**.**
In view of Proposition 3.9, it is natural to wonder whether the class of frequently hypercyclic operators is also Borel in . Let be a countable basis of open sets for . An operator is frequently hypercyclic if and only if
[TABLE]
and since the relation defined by the expression between brackets is Borel in , we deduce that is a (aka analytic) subset of . It is quite tempting to conjecture that it is in fact non-Borel in .
3.2. Some non-Borel sets in
We conclude this section by showing that some natural classes of operators defined by dynamical properties are non-Borel in the space , for any .
If is any class of Hilbert space operators, we will say that an operator has a restriction in if there is a closed -invariant (infinite-dimensional) subspace of such that the operator induced by on belongs to . We denote by the class of all operators admitting a restriction in .
Proposition 3.23**.**
Let be a class of operators on such that G-ERGHC. For any , the set is non-Borel in .
Démonstration.
Recall that for every compact subset of , denotes the Kalisch operator associated to (this notation has been introduced in the proof of Proposition 3.4). For any compact subset of , we denote by the set of all infinite compact subsets of .
Fact 3.24**.**
Let be a compact subset of . There exists a sequence of Borel maps from into such that is an orthonormal basis of for every .
Proof of Fact 3.24.
By the Kuratowski-Ryll-Nardzewski Selection Theorem (see [37, Section 12.C]), there exists a sequence of Borel maps such that is a dense subset of for every . If belongs to , there exists a sequence of distinct elements of such that is dense in . Indeed, it suffices to set and, for every , where . Then the maps , , are Borel on , and by construction the elements , , are pairwise distinct and form a dense subset of . Applying the Gram-Schmidt orthonormalization process to the sequence , which is linearly independent, we get the required orthonormal basis of . ∎
The key step in the proof of Proposition 3.23 is the following result of Waterman [52].
Fact 3.25**.**
If is a compact subset of , the operator admits spectral synthesis, which means that every closed -invariant subspace of is spanned by the eigenvectors belonging to it. In other words, any invariant subspace for has the form for some compact set .
We are now ready to prove Proposition 3.23. Let us fix a non-trivial arc such that . As in the proof of Proposition 3.4, let be an orthonormal basis of , and for any , denote by the unitary operator sending to for all , where is given by Fact 3.24. Then define a map by setting
[TABLE]
Proceeding as in the proof of Proposition 3.4, and using the fact that the maps , , are Borel on , we deduce that the map is Borel from into .
If is uncountable, it admits a perfect subset , so the restriction of to the invariant subspace (which is nothing else but ) is ergodic in the Gaussian sense. Thus belongs to \widehat{\hbox{G-}ERG}(\mathcal{H})\subseteq\widehat{\Gamma}(\mathcal{H}). On the other hand, if is countable then, by Fact 3.25, has no hypercyclic restriction. Hence belongs to . So we have proved that
[TABLE]
Since is notoriously non-Borel in (see [37, Section 27.B]) and hence also in because the latter is Borel in , this concludes the proof. ∎
Remark 3.26**.**
The same proof shows that , the family of all operators admitting a restriction which is mixing in the Gaussian sense, is non-Borel in . Instead of working with the class of countable subsets of , one has to consider the class of compact subsets of which are sets of extended uniqueness, also called -sets. By definition, a compact set is a -set if it is negligible for every probability measure on whose Fourier coefficients vanish at infinity. It follows from the main result of [8] that if is not a -set, then the Kalisch operator has a restriction which is mixing in the Gaussian sense, whereas if is a -set, is not mixing in the Gaussian sense. Moreover, it is well-known that the class is non-Borel in for any non-trivial arc (see [38, Th. VIII.2.6]). Using these facts, one can prove the aforementioned result in exactly the same way as above: the key identity
[TABLE]
at the end of the proof of Proposition 3.23 is replaced by
[TABLE]
Remark 3.27**.**
Let be a class of operators of with the following two properties:
- (i)
; 2. (ii)
if is any countable compact subset of , the Kalisch operator does not belong to .
Exactly the same proof as that of Proposition 3.23 shows that the following statement holds true: if is a class of operators on such that then, for any , the set is non-Borel in .
Our aim is now to prove the following result:
Proposition 3.28**.**
For any , the class of distributionally chaotic operators in is non-Borel in .
Note that this is in strong contrast with Proposition 2.37, according to which the class of densely distrobutionally chaotic operators in is .
The proof of Proposition 3.28 relies on the following observation:
Fact 3.29**.**
An operator is distributionally chaotic if and only if it has a restriction which is densely distributionally chaotic.
Proof of Fact 3.29.
If there exists a closed subspace of such that is densely distributionally chaotic, admits in particular a distributionally irregular vector belonging to , and hence is distributionally chaotic. Conversely, assume that is distributionally chaotic, and let be a distributionally irregular vector. Let be the closed -invariant subspace spanned by the orbit of , and denote by the operator induced by on . As the vector has a distributionally unbounded orbit under the action of (*i.e. * as along a subset of , with ), admits by [10, Prop. 8] a comeager set of vectors with distributionally unbounded orbit. Also, the orbit of under the action of is distributionally near to [math] (i.e. as along a subset of , with ). It follows that as along for every vector belonging to the linear span of the vectors , . Hence admits a dense set of vectors whose orbit is distributionally near to [math]. By [10, Prop. 9], it follows that the set of vectors whose orbit under the action of is distributionally near to [math] is comeager in . By the Baire Category Theorem, we conclude that admits a comeager set of distributionally irregular vectors, i.e. is densely distributionally chaotic. ∎
We will also need the following characterizations of (densely) distributionally chaotic Kalisch operators:
Fact 3.30**.**
Let be a compact subset of , and let be the associated Kalisch operator on .
- (1)
The operator is densely distributionally chaotic if and only if is a perfect set. 2. (2)
The operator is distributionally chaotic if and only if is uncountable.
Proof of Fact 3.30.
Let us start by proving assertion . If is a perfect set then is an ergodic operator, so that it is in particular densely distributionally chaotic by [29]. Conversely, suppose that is not a perfect set, and let be an isolated point of . Then the eigenfunction does not belong to , and can be written as a topological direct sum . Hence, there exists a non-zero vector of with and , and a bounded projection of onto , such that for every . It follows that for every , so that . Hence for every . Thus, we see that the orbit under of any vector with is bounded away from [math], and hence that no such vector can be distributionally irregular. So is not densely distributionally chaotic.
The proof of assertion relies on Facts 3.25 and 3.29 above. By Fact 3.29, is distributionally chaotic if and only if it admits a restriction which is densely distributionally chaotic. Since, by Fact 3.25, any such restriction is of the form , where is a compact subset of , it follows from assertion that is distributionally chaotic if and only if contains a perfect set, which happens exactly when is uncountable. This proves . ∎
We are now ready for the proof of Proposition 3.28.
Proof of Proposition 3.28.
Consider the class . This class satisfies the assumptions of Remark 3.27: is contained in by a result of [29], and it follows from Fact 3.30 that whenever is a countable subset of , the Kalisch operator is not (densely) distributionally chaotic. Since , we conclude that for any , is non-Borel in . But by Fact 3.29, and hence is non-Borel in . ∎
Corollary 3.31**.**
For any , the set of operators in which are distributionally chaotic but not densely distributionally chaotic is non-Borel in .
Démonstration.
Since is a subset of by Proposition 2.37, it is in particular Borel. So one immediately deduces from Proposition 3.28 that is non-Borel in . ∎
Remark 3.32**.**
Corollary 3.31 formally yields the existence of distributionally chaotic operators which are not densely distributionally chaotic. But the existence of such operators can of course be deduced directly from Proposition 3.30: a Kalisch operator is distributionally chaotic but not densely distributionally chaotic if and only if is uncountable but not perfect.
4. Ergodicity for upper-triangular operators
4.1. Definitions and setting
Let us fix an orthonormal basis of the complex separable infinite-dimensional Hilbert space . We will denote by the linear subspace of consisting of finitely supported vectors with respect to the basis .
In this subsection, we concentrate ourselves on the study of the typical properties of operators on which are upper-triangular with respect to the basis , with diagonal coefficients of modulus and pairwise distinct. This class of operators will be denoted by :
[TABLE]
For each and , we write . We also define
[TABLE]
and
[TABLE]
Recall that if is a finite or infinite family of elements of with for every , it is said to be rationally independent if for any finite subset of , the family consists of -independent numbers.
For any , we denote by , , and the set of operators in which belong to , , and respectively. It is not difficult to check that these sets are in . So we may state:
Fact 4.1**.**
For any , the sets , , and are Polish spaces when endowed with the topology SOT.
Note that (by the Gram-Schmidt orthonormalization process) any bounded operator on with spanning unimodular eigenvectors associated to distinct eigenvalues is upper triangular with respect to some orthonormal basis , and belongs to the associated class . Hence, given the importance in linear dynamics of unimodular eigenvectors, it is quite natural to investigate typical properties of elements of the spaces , . Some interesting and unexpected phenomena occur in this setting, which we will describe shortly. Meanwhile, we state for future reference two elementary facts concerning the SOT-continuity of certain maps naturally associated to operators in .
When belongs to , there exists for each a unique vector with such that . We denote this vector by , so that . We also denote by the normalized eigenvector of . Here is a useful fact concerning the functions and :
Fact 4.2**.**
For each , the functions and are continuous from into and respectively.
Démonstration.
The continuity of the function is obvious from the definition of . As to the function , the fact that implies that
[TABLE]
for every . A straightforward induction then shows that the scalar functions , , are SOT-continuous on , from which Fact 4.2 follows. ∎
Fact 4.3**.**
Let , and choose such that . For any , can be written as , where the functions , , are continuous from into .
Démonstration.
This follows directly from Cramer’s formulas and the continuity of the maps , . ∎
4.2. Perfect spanning is typical
The following result shows that within any class , , the perfect spanning property (aka ergodicity in the Gaussian sense) is a typical property with respect to the topology SOT.
Proposition 4.4**.**
For every , the set is comeager in . Consequently, , and are all comeager in .
Démonstration.
We first recall the following key fact, which goes back to [27, Th. 4.1] (see also [29, Prop. 6.1]).
Fact 4.5**.**
An operator has a perfectly spanning set of unimodular eigenvectors as soon as the following property is satisfied: for every and every , there exists with and such that .
From Fact 4.5, we deduce that the set
[TABLE]
is contained in . Moreover, it follows from Fact 4.2 that is also a subset of . So we just have to prove that is dense in ; and for this, it is enough to show that each open set
[TABLE]
is dense in .
Let us fix and . We have to show that for any , there exists an operator such that is arbitrarily close to for every . Without loss of generality, we assume that . Without loss of generality, we can suppose also that . Indeed, if , consider the function defined on the compact set , where is the diagonal operator given by the diagonal coefficients of . The function is convex on , so that its maximum is attained at either [math] or . Since , for every . If is sufficiently close to , the operator is upper-triangular with respect to , has the same diagonal coefficients as , satisfies , and is as close as we wish to for the SOT topology. We thus suppose to begin with that .
We set, for every , , and denote by the orthogonal projection of onto .
For every , we denote by the -th diagonal coefficient of : . We recall that . Let us denote by the set of all sequences with such that all the elements , , are pairwise distinct and distinct from , and as . For any element of and any unilateral weight sequence , we consider the operator on defined by
[TABLE]
Note that since , for every provided is small enough. Hence in this case belongs to . Moreover, we have for every . Let us now fix a weight sequence such that as and is small enough. We are going to show that belongs to provided the sequence is well-chosen. This will prove the density of in .
We first note the following fact concerning the eigenvalues of the operator .
Fact 4.6**.**
For any , the only eigenvalues of are , the terms of the sequence , and possibly the point . Moreover, for any , the eigenspace associated to is spanned by the vector .
Proof of Fact 4.6.
If is any complex number, solving formally the equation with , yields that
[TABLE]
Since and as , we infer from these equations that there is no solution to these equations if does not belong to the set . This proves the first part of Fact 4.6.
For the same reasons combined with the fact that the are pairwise distinct and distinct from , we also observe that if a vector satisfies for some , then for all . Hence is contained in for every , which proves the second part of Fact 4.6 since belongs to and the eigenvalues of are all distinct. ∎
The key point is now the following observation. Recall that are fixed, that , and that we are especially interested in . Recall also that .
Fact 4.7**.**
There exists a positive constant such that for any ,
[TABLE]
Démonstration.
For any , the restriction of the operator to is equal to , which does not depend on . Also, , and the eigenvalues of are distinct. These observations imply that there exist two positive constants and such that the following two properties hold true:
[TABLE]
and
[TABLE]
Now, a simple computation yields that
[TABLE]
where satisfies the equation . Since does not belong to the set , we may write , and implies that
[TABLE]
On the other hand, , so that by (8)
[TABLE]
Hence
[TABLE]
and
[TABLE]
Setting yields the desired inequality. ∎
We now come back to our main proof: recall that our aim is to show that belongs to for some suitable choice of . By Fact 4.7, we have
[TABLE]
We now choose such that is so close to that the quantity on the right hand side of the inequality (11) is less that , where \eta_{k}=\min\bigl{(}1,(2\|u_{k}(A_{\boldsymbol{\lambda},\boldsymbol{\omega}})\|)^{-1}\bigr{)}. Note that does not depend on since the restriction of to does not. With this choice of , there exists a scalar such that . Since and , we have , so that . It follows that , where , and this shows that belongs to .
Thus we have proved that is indeed dense in for every , which concludes the proof of Proposition 4.4. ∎
Remark 4.8**.**
It is natural to wonder whether the class of chaotic operators is also comeager in . This does not look quite clear from the above proof.
4.3. Ergodicity vs ergodicity in the Gaussian sense
Let be as usual a fixed orthonormal basis of . In this subsection, we focus on special operators of the associated class of upper-triangular operators with respect to , which are the sum of a diagonal operator with respect to the basis and a backward weighted shift operator with respect to this same basis. More precisely, we introduce the following notation: we denote by the set of sequences of pairwise distinct complex numbers of modulus such that tends to as tends to infinity, and for every , by the set of all weight sequences such that for every . We also set .
We endow the set with the topology induced by . Since is contained in the separable closed subspace of consisting of all convergent sequences, and since is easily seen to be a subset of , it follows that is a Polish space. As for the spaces , , we endow them with the product topology. Each being a subset of , it is thus a Polish space as well.
To each pair , we associate the operator defined by , where is the diagonal operator with diagonal coefficients , , associated to the basis , and is the backward shift operator with respect to with weights , . Each operator belongs to .
In this subsection, our aim is to investigate the properties of the operator for a typical choice of parameters . For every , we consider the following two sets of parameters:
[TABLE]
and
[TABLE]
where denotes the point spectrum (i.e. the set of eigenvalues) of an operator . The operators belonging to are those which have the smallest possible set of eigenvalues among the operators , . In particular, they have countable unimodular point spectrum.
Our main result can now be stated as follows:
Theorem 4.9**.**
For any , the two sets and are comeager in .
As an immediate consequence, we obtain
Corollary 4.10**.**
There exist ergodic operators on , of the form , , which have countable unimodular point spectrum. In particular, these operators are ergodic but not ergodic in the Gaussian sense.
Remark 4.11**.**
Corollary 4.10 provides examples of hypercyclic operators on a Hilbert space with a spanning set of unimodular eigenvectors but only countably many unimodular eigenvalues. The question of the existence of such operators was first raised by Flytzanis in [22], and answered recently in [39]. Indeed, the chaotic non-frequently hypercyclic operators constructed there have only countably many eigenvalues, which are all roots of unity, and the associated eigenvectors span the space. Corollary 4.10 strengthens this result by showing that ergodic counterexamples to Flytzanis’ conjecture exist, and that such counterexamples are in some sense much less exotic than suggested by the rather technical construction of [39].
Remark 4.12**.**
Since all the operators we are considering here have infinitely many eigenvalues, Corollary 4.10 leaves open the question of the existence of ergodic operators on without any eigenvalue at all. Such operators are known to exist on the Banach space of continuous functions on vanishing at the point [math]. Indeed, the Kalisch operator defined by
[TABLE]
when considered as acting on , is ergodic in the Gaussian sense but does not admit any unimodular eigenvalue; see [5] or [7, Section 5.5.4] for details. On the other hand, an operator on a Hilbert space which is ergodic in the Gaussian sense definitely has a lot of unimodular eigenvalues (recall that ); but a general ergodic operator might possibly have no eigenvalue at all.
We now turn to the proof of Theorem 4.9. We first state a simple fact, in which a complete description of the unimodular eigenvectors of the operators is given.
Fact 4.13**.**
Fix , and . Then is an eigenvalue of if and only if the vector
[TABLE]
is a well-defined vector of . In this case, =.
Démonstration.
It suffices to solve formally the equation in . ∎
For every ,
[TABLE]
is thus an eigenvector of associated to the eigenvalue , and hence is proportional to .
The next lemma provides necessary conditions for the point spectrum of to be either “maximal” or “minimal”. For any element of , we denote by the radius of convergence of the series
[TABLE]
Lemma 4.14**.**
Let be an element of .
- (1)
If , the map is well-defined and analytic on a neighborhood of the unit circle . Consequently, for every . 2. (2)
If , the eigenvalues of are contained in the set .
Démonstration.
The first part of assertion (1) is a direct consequence of Fact 4.13: since , the series defining is convergent in for every belonging to the open disk , and the map is analytic there. In particular, this map is analytic in a neighborhood of . Since the sequence has an accumulation point in for every , it follows from the analyticity of that the closed linear span of the vectors , , coincides with the closed linear span of all the eigenvectors , . Since is proportional to and since the vectors , , span a dense subspace of , this proves the second part of assertion (1).
As to assertion (2), suppose that . Let us show that the series defining does not converge when does not belong to . So let fix such a complex number . Since tends to as tends to infinity, there exists a number such that for every . Hence
[TABLE]
Since , it follows that
[TABLE]
and hence is not defined as a vector of . ∎
After these preliminary facts, we now state a proposition which will be the key step to prove that ergodic elements are typical in for any . We postpone the proof of Proposition 4.15 to the end of this section, and explain first how Theorem 4.9 may be deduced from it. In what follows, we fix . For any , any open set in and any , we set
[TABLE]
Note that is an open subset of : this follows from the continuity of the map from into , combined with Fact 4.2.
Proposition 4.15**.**
Let be a non-empty open set in , and let . If is such that , then the open set is dense in .
Proposition 4.15 will allow us to apply the following ergodicity criterion proved in [29, Cor. 5.5].
Lemma 4.16**.**
Let be a separable Banach space, and let . Assume that for any open set of with , any neighborhood of [math], and any , there exists a -invariant probability measure on with compact support such that and . Then is ergodic.
The link with Proposition 4.15 is perhaps not quite obvious at first sight. It is provided by the following fact.
Fact 4.17**.**
Let , let be a non-empty open set in , let , and let be a neighborhood of [math] in . Let also , and assume that the eigenvalues of are rationally independent. Finally, let be complex numbers and assume that the vector belongs to . If the quantity is small enough, there exists a compactly supported -invariant measure on such that
, where , and
.
Démonstration.
We only sketch the proof, since the argument is already essentially given in [29]. Let be a sequence of independent random Steinhaus variables defined on some standard probability space , and let be the measure on given by the distribution of the -valued random variable . This measure , which may be called the Steinhaus measure associated with the vector , is -invariant and has compact support. As observed in [29, Fact 5.16], the independence of the unimodular numbers and the condition imply that . Also, we have by orthogonality of the Steinhaus variables , , that
[TABLE]
It then follows from Markov’s inequality that provided that the quantity is sufficiently small. ∎
Proof of Theorem 4.9.
The proofs of the two parts of Theorem 4.9 are completely independent of each other. Recall that the sets and are defined at the beginning of Section 4.3, and that we assume that .
Part 1. The set is comeager in .
Let be a countable basis of non-empty open subsets for . For each , define the subset of
[TABLE]
and
[TABLE]
Claim 4.18**.**
The set is a dense subset of .
Proof of Claim 4.18.
The fact that is in is immediate. As to its density, we first observe that since , Proposition 4.15 implies that is dense in as soon as for sufficiently large. It follows that the set is dense in . Since this set is also clearly in , it is thus comeager in . The Kuratowski-Ulam theorem then implies that the set is comeager and hence dense in . ∎
It is now easy to show that the set is comeager in . Indeed, it follows from Lemma 4.16 and Fact 4.17 that is ergodic as soon as belongs to and the terms , , of the sequence are rationally independent. Hence the set
[TABLE]
is contained in . Since the conditions that the numbers should be rationally independent is easily seen to define a dense subset of , Claim 4.18 implies that is a dense subset of . Hence is comeager in .
Part 2. The set is comeager in .
The argument here is much simpler, and relies solely on part of Lemma 4.14. Observe that the set of all such that is in , and that this set is dense in because it contains all sequences such that tends to [math] as tends to infinity. Hence the set is and dense in , and the claim follows. ∎
We now turn to the proof of Proposition 4.15. Again, it relies on some arguments used in [29], more precisely in the proof of [29, Th. 5.12].
Proof of Proposition 4.15.
Let us fix and . We are looking for an element of such that .
Writing as , we fix such that for every . By assertion (1) of Lemma 4.14, the linear span of the vectors , , is dense in , so there exists an index and complex numbers such that the vector
[TABLE]
We now proceed as in [27] or [29], and replace each coefficient by a certain sum of scalars of the form with sufficiently small. More precisely, we define
[TABLE]
for every and every , where is an integer so large that
[TABLE]
Observe that by construction, so that we have
[TABLE]
The next step in the proof is to define with , in such a way that each vector , , can be approximated by the eigenvector of associated to the eigenvalue , and the , , are all distinct. The sequence is defined as follows: we keep for every and every , so that in particular as . For and , we choose extremely close to , in such a way that these new coefficients are all distinct and distinct from all the . Since we already know that as , the sequence defined in this way belongs to . We can certainly ensure that for all and , and thus that
[TABLE]
since . All the remaining coefficients of and coincide, and hence .
Let us now show that the quantities
[TABLE]
where and , can be made as small as we wish, provided each coefficient is close enough to . We consider separately two cases.
-
Assume that and . In this case by definition, so that .
-
Assume now that and . In this case, we use the explicit expression of the eigenvectors provided by Fact 4.13 and write
[TABLE]
Fix an arbitrarily small number . The integers and being fixed, and the numbers and being by definition equal for every , the norm of the first sum in the above expression can be made less that , provided that the difference is sufficiently small. As for the second term, observe that belongs to the set for every , so that the term always appears in the product . Thus if is sufficiently small, the norm of the second term is less than too. Hence one can ensure that for all and .
So we have proved that for any , one can construct such that and for every . If is now chosen small enough, the two conditions
[TABLE]
simultaneously hold true, by (13) and (12) respectively. Remembering that each vector is proportional to , we eventually obtain that there exist complex coefficients , , such that
[TABLE]
Hence belongs to , and this concludes the proof of Proposition 4.15. ∎
Remark 4.19**.**
In order to show that the ergodic operators are comeager in , we have used in a crucial way the fact that the terms , , of the sequences involved in the proof are rationally independent. This is not so surprising in view, for instance, of [29, Fact 5.16]. Yet the role of independence in these issues remains rather mysterious. We develop this a little bit in the next subsection.
4.4. Additional remarks
In this subsection, we present a short discussion of some questions motivated by the results obtained above, as well as some further results concerning dynamical properties of the “diagonal plus shift” operators .
4.4.1. Some natural questions
Two such questions concerning the existence of operators on a Hilbert space with particular ergodic-theoretic-like properties remain unanswered at this stage of our work. The first one is
Question 4.20**.**
Do there exist -frequently hypercyclic operators on which are not frequently hypercyclic?
It was proved by Bayart and Rusza in [9] that such operators do exist on the space , but the question was left open for Hilbert (or even reflexive) spaces. In the light of the discussion carried out in Section 3 of this paper, it seems natural to conjecture that the two classes and should have different descriptive complexity, and hence should be distinct; but we have been unable to solve the question using this approach. The second question runs as follows:
Question 4.21**.**
Do there exist frequently hypercyclic operators on which are not ergodic?
This question comes from [29], where it is proved that frequently hypercyclic non-ergodic operators do exist on the space . The proof of this result again relies on a construction of [9]: it is proved in [29] that the frequently hypercyclic bilateral weighted shifts on defined in [9] satisfy , and thus cannot be ergodic. For some reasons, we found it rather tempting to try to attack this question by Baire category methods; but again, we did not succeed in this way.
We do solve Questions 4.20 and 4.21 in Section 6 below, but using widely different methods. The operators we will use are generalizations of those introduced by the third named author in [39] in order to solve the question of the existence of a chaotic non-frequently hypercyclic operator. The two main results we will obtain are that indeed, there exist -frequently hypercyclic operators on which are not frequently hypercyclic, as well as frequently hypercyclic operators which are not ergodic. All our examples turn out to be chaotic; and we will complement these results by showing that there also exist on operators which are chaotic and topologically mixing but not -frequently hypercyclic.
In another direction, the role of rational independence in all that concerns the links between unimodular eigenvalues and ergodicity properties of operators needs to be clarified. To be a little more specific, let us consider the class of operators in whose diagonal coefficients are rationally independent. Rather surprisingly, it seems that very little is known concerning such operators. For example, to our knowledge the following question is open:
Question 4.22**.**
Let be a hypercyclic operator belonging to . Is necessarily ergodic? frequently hypercyclic? -frequently hypercyclic?
The only currently known examples of hypercyclic operators with spanning unimodular eigenvectors which are not frequently hypercyclic or not ergodic are those constructed in [39] and the ones that will be considered in Section 6. As already mentioned, all these operators are chaotic. Moreover, for many of these operators, the only unimodular eigenvalues are roots of unity and for some of those, each eigenvalue has multiplicity one. So these operators belong to for some suitably chosen orthonormal basis , but not to for any basis . A positive answer to Question 4.22 would show that this is not accidental, i.e. that there must be a strong amount of dependence between the eigenvalues of any of the operators constructed in [39] or in Section 6 of the present paper.
As a matter of fact, Question 4.22 seems to be open even for the operators considered in Subsection 4.3. Even more prosaically, it seems quite desirable (and perhaps not too difficult) to determine when exactly an operator of the form is hypercyclic.
4.4.2. More on the operators
We finish this section by collecting some simple facts that we do know concerning operators of the form . Let us introduce the following notations.
If is a unilateral weight sequence, we set and
[TABLE]
Also, using the same notation as in Section 4.3 above, we denote by the radius of convergence of the series :
[TABLE]
To each , we associate polynomials , , defined as follows: and
[TABLE]
Proposition 4.23**.**
Let be a unilateral weight sequence, and let . Then the following facts hold true.
- (0)
A complex number is an eigenvalue of if and only if
[TABLE]
In this case, where
[TABLE] 2. (1)
If , then . 3. (2)
If (e.g. if as ), then . More generally, is contained in the set , where denotes the spectral radius of .
Assume now that belongs to , i.e. that the elements , , are pairwise distinct and that as .
- (3)
If , then contains and is contained in . 2. (4)
If , then is ergodic in the Gaussian sense. Moreover, the map is analytic on the open disk . 3. (5)
If , then is not hypercyclic. 4. (6)
If , let be the largest integer such that . If
[TABLE]
then is not ergodic in the Gaussian sense.
Démonstration.
(0) This is a simple computation which has already been carried out in the proof of Fact 4.13.
(1) If does not belong to the closure of , then , and thus for all . If , assertion (0) implies that is not an eigenvalue of .
(2) Let . We have , where is defined by for every . If , then is invertible and can be written as
[TABLE]
where . Moreover, since , the spectral radius of is at most . It follows that is invertible as soon as .
(3) We already know by assertion (0) that contains . Now, let belong to the open disk , and choose such that . Since , there exists such that for all . It follows that there exists a constant such that for every , so that is well-defined and analytic on .
Conversely, assume that does not belong to the set . Since , one can find such that for sufficiently large; and since for all , it follows that there exists a constant such that for every . Hence is not well-defined.
(4) By assertion (3), the map is analytic on the disk ; so it is enough to show that the vectors , , span a dense subspace of . Let be a vector of which is orthogonal to all the vectors , . We thus have
[TABLE]
The series above is convergent on the disk , which contains all the points by assumption. So
[TABLE]
Since whenever and , it follows that for every , so that .
(5) Let be such that . Then is an isolated eigenvalue of by assertion (3), which is easily seen not to belong to the essential spectrum of . It then follows from [34] that cannot be hypercyclic.
(6) By assertion (1), we can assume that , since otherwise the point spectrum of is countable, and is certainly not ergodic in the Gaussian sense in this case. It suffices to show that the unimodular eigenvectors of are not perfectly spanning. Since is a finite set by assertion (3), it is in turn enough to show that the vectors , , do not span a dense subspace of . So we have to find coefficients , , with the property that
[TABLE]
with the additional requirement that
[TABLE]
We define the coefficients by setting for every , and
[TABLE]
Then condition (14) is satisfied by assumption. Moreover, the coefficients have been defined in such a way that
[TABLE]
Since and as , it follows that for every such that . This completes the proof of assertion (6). ∎
5. Periodic points at the service of hypercyclicity
In this section, we depart from our standing assumption and work in the general context of Banach spaces over or , not restricting ourselves to the Hilbertian setting. Actually, most of our results hold in the framework of arbitrary Polish topological vector spaces, as should be clear from the proofs.
Let thus be a real or complex separable Banach space. Our aim is to obtain general sufficient conditions for an operator to be frequently hypercyclic or -frequently hypercyclic. These criteria are rather different from the classical ones, since they involve the periodic points of the operator . They will greatly contribute to simplify some proofs in the next section, and they might hopefully be useful elsewhere also. For any bounded operator on , we denote by the set of its periodic points, and by , or simply , the period of a periodic point of .
5.1. Precompact orbits and topological mixing
We start by giving a very simple criterion for an operator to be topologically mixing, which involves the points of with precompact orbit under the action of . We denote by the set of points with precompact orbit.
Proposition 5.1**.**
Let . Assume that is dense in and that for every neighborhood of [math] in and every non-empty open subset of , the set is cofinite. Then is topologically mixing.
Démonstration.
It suffices to show that for any vectors , , and any , the set \mathcal{N}_{T}\bigl{(}B(y,\varepsilon),B(x,\varepsilon)\bigr{)} is cofinite. Since is a vector with precompact orbit, there exists a finite subset of such that any point of lies within distance less than of the set . In other words, there exists for any an index such that .
By assumption, each set \mathcal{N}_{T}\bigl{(}B(0,\varepsilon),B(x-y_{i},\varepsilon/2)\bigr{)} is cofinite. Hence, one can find an integer such that for every and every , there exists such that
[TABLE]
For every , we then have
[TABLE]
Hence belongs to and belongs to , which shows that every integer lies in the set \mathcal{N}_{T}\bigl{(}(B(y,\varepsilon),B(x,\varepsilon)\bigr{)}. ∎
Since periodic points have a finite orbit, Proposition 5.1 immediately implies the following result.
Corollary 5.2**.**
Let . If is dense in and if for every neighborhood of [math] in and every non-empty open subset of , the set is cofinite, then is chaotic and topologically mixing.
We also obtain in a similar fashion.
Corollary 5.3**.**
Let . If there exists a dense set of points such that as and if for every neighborhood of [math] in and every non-empty open subset of , the set is cofinite, then is topologically mixing.
Finally, the linearity assumption on the system implies the following strengthened version of Proposition 5.1.
Corollary 5.4**.**
Let . Assume that is dense in and that there exists a subset of with dense linear span such that the following property holds true: for every and every , there exists such that for every , one can find such that and . Then is topologically mixing.
Démonstration.
Let be a non-empty open set. Since spans a dense subspace of , there exist and with and for every such that is contained in . Moreover, by assumption, there exists such that for every , every , one can find such that
[TABLE]
The vector then satisfies and , and the desired result follows from Proposition 5.1. ∎
Remark 5.5**.**
Unlike in Corollary 5.8 below, it is not possible, in Proposition 5.1, to replace the assumption that all the sets are cofinite ( and non-empty open subsets of with ) by the assumption that all the sets are cofinite ( and non-empty open subsets of with ), even if one assumes additionally that is hypercyclic. Indeed, every unilateral weighted backward shift has a dense set of points with finite orbit and is such that is cofinite for every neighborhood of [math] and every non-empty open set ; but there exist hypercyclic weighted shifts which are not topologically mixing.
5.2. Uniform recurrence and topological weak mixing
In this subsection, we give a simple criterion for an operator to be weakly topologically mixing. It involves the uniformly recurrent points of . Recall that a point is said to be recurrent for if, for any neighborhood of , the set is non-empty (or, equivalently, infinite). A point is said to be uniformly recurrent for if, for any neighborhood of , the set has bounded gaps. For example, every periodic point is obviously uniformly recurrent. The following more general fact will be useful. Recall first that a compact, -invariant subset of is said to be minimal if it has no proper (non-empty) closed -invariant subset. For example, if is a periodic point of with period , then is minimal. Recall also that we denote by the set of all unimodular eigenvectors of .
Fact 5.6**.**
Let . If is a compact minimal -invariant subset of , then every point of is uniformly recurrent for . In particular, every point is uniformly recurrent for .
Démonstration.
The first part of the statement is well-known (and has nothing to do with linearity; see e.g. [23, Theorem 1.15]). As for the second part, it is enough to show that if belongs to , then its closed -orbit is a compact minimal -invariant set.
Write as , where for every , for some . Let be the closed subgroup of generated by the -tuple . Then , so that is compact (and, of course, - invariant). So we just have to check that is minimal for the action of . Let be arbitrary. We have to show that any point can be approximated as close as we wish by points of the form , . Write and as and respectively, where and belong to . For any integer , we have . Since the -tuples , , are dense in (this follows from the fact that since the group is compact, the closed semigroup generated by is in fact a group), there exists an integer such that is as close as we wish to . Then is as close as we wish to , as required. ∎
We can now state our criterion for topological weak mixing.
Proposition 5.7**.**
Let , and let be a -invariant subset of consisting of uniformly recurrent points. Assume that for every non-empty open subset of and every neighborhood of [math] in , one can find a vector and an integer such that and . Then is topologically weakly mixing and is dense in .
Démonstration.
That is dense in is clear from the assumption since is -invariant. In order to prove that is topologically weakly mixing, we are going to show that is non-empty for any non-empty open subsets of and any neighborhood of [math] in . The so-called three open sets condition (see for example [7, Ch. 4]) will then imply that is topologically weakly mixing.
Let us first show that for any neighborhood of [math] and any open set , the set has bounded gaps. Choose a vector and an integer such that belongs to and belongs to , and then an open neighborhood of such that and . Since is uniformly recurrent, the set has bounded gaps. Then has bounded gaps as well, and the result follows since .
We show next that is non-empty for any non-empty open subset of and any neighborhood of [math] in . By assumption, one can find a point and an integer such that is a (uniformly) recurrent point of and lies in . Since is recurrent, one can find an integer such that lies in . Then belongs to , and hence is non-empty.
It is now easy to conclude the proof: all the sets have bounded gaps, and all the sets contain arbitrarily large intervals, because they are all non-empty and the sets are neighborhoods of the fixed point [math]. Hence any set of the form meets any other set of the form . ∎
As a first consequence of Proposition 5.7, we show that for operators having a dense set of uniformly recurrent points, topological weak mixing is equivalent to some formally much weaker “transitivity-like” properties.
Corollary 5.8**.**
Let , and assume that uniformly recurrent points of are dense in . The following assertions are then equivalent:
- (a)
* is topologically weakly mixing;* 2. (b)
for every non-empty open subset of and every neighborhood of [math] in , the set is non-empty; 3. (c)
for every non-empty open subset of and every neighborhood of [math] in , the set is non-empty.
Démonstration.
Obviously, (a) implies both (b) and (c). Moreover, since the uniformly recurrent points of are assumed to be dense in , it follows at once from Proposition 5.7 (taking as the set of all uniformly recurrent points for ) that (b) implies (a). So it remains to show that (c) implies (b).
Suppose that (c) holds true, and let and be two non-empty open subsets of with . By (c) and since uniformly recurrent points of are dense in , one can find a uniformly recurrent point and an integer such that belongs to . Since is in particular recurrent, one can find such that belongs to . Then lies in , so that is non-empty. ∎
As another consequence of Proposition 5.7, we now state a criterion for topological weak mixing which formally resembles the ergodicity criterion stated as Lemma 4.16 above.
Corollary 5.9**.**
Let . Assume that for each non-empty open subset of such that and each neighborhood of [math], one can find a minimal -invariant compact set such that is non-empty. Then is topologically weakly mixing.
Démonstration.
By Fact 5.6, every point in the above compact set is uniformly recurrent for . We apply Proposition 5.7, taking as the set of all uniformly recurrent points for . Let be a non-empty open subset of . Applying the assumption of Corollary 5.9 to , we immediately see that the assumption of Proposition 5.7 is satisfied, and hence that is topologically weakly mixing. ∎
Corollary 5.10**.**
Let , where is a complex Banach space. Assume that for every and , there exist and such that and . Then is topologically weakly mixing and is dense in .
Démonstration.
This follows from Fact 5.6 and Proposition 5.7 applied with . ∎
Our last corollary is the result on which we will elaborate to state our -frequent hypercyclicity and frequent hypercyclicity criteria.
Corollary 5.11**.**
Let . Assume that for every and , there exist and such that and . Then is chaotic.
Démonstration.
Since periodic points are uniformly recurrent, exactly the same proof as that of the previous corollary shows that is topologically weakly mixing and has a dense set of periodic points. ∎
Corollary 5.11 can also be proved by a more constructive argument. Since the flexibility of this alternative proof will prove extremely important below for the proofs of some (-) frequent hypercyclicity criteria, we present it here.
Direct proof of Corollary 5.11.
Before starting the proof, we note that the assumption of Corollary 5.11 obviously implies that and that is dense in . The first step of the proof is to observe that the assumption of Corollary 5.11 can be reinforced as follows:
Fact 5.12**.**
Under the assumption of Corollary 5.11, the following property holds true: for any , any , and any integers , there exist and with such that and .
Proof of Fact 5.12.
Assuming (as we may) that is non-zero, choose and such that and , where has to be specified. Then , so that It follows that if is small enough, then . Let then be an integer such that is equal to , and set . Then , so if is small enough. Also, , so that . ∎
Let now be a dense sequence of vectors of belonging to . A straightforward induction shows that there exists a sequence of elements of , as well as a strictly increasing sequence of integers such that, for every ,
- (i)
; 2. (ii)
; 3. (iii)
is a multiple of the period of the vector .
Now, set
[TABLE]
which is well-defined by (i). Let us now show that is a hypercyclic vector for . For every , we have \displaystyle T^{n_{j}}\Big{(}\sum_{i<j}z_{i}\Big{)}=\sum_{i<j}z_{i} by (iii), so that
[TABLE]
It follows that
[TABLE]
This terminates our direct proof of Corollary 5.11. ∎
To conclude this subsection, we prove a slightly stronger version of Corollary 5.11.
Corollary 5.13**.**
Let be a subset of with dense linear span. Suppose that for every and every , there exist and such that and . Then is chaotic.
Démonstration.
It is suffices to show that the assumption of Corollary 5.13 can be extended from vectors of to arbitrary vectors of : once this is done, Corollary 5.11 applies.
Since the linear span of is dense in , it suffices to consider vectors of the form , where and for every . An induction on , , allows us to construct, using (an obvious modification of) Fact 5.12, vectors and integers , , with the following properties:
- (i)
; 2. (ii)
; 3. (iii)
is a multiple of .
Now, set : this is clearly a periodic vector for , with by (i). Set also . We have
[TABLE]
This concludes the proof of Corollary 5.13. ∎
In the next two subsections, we will give two “variations” of Corollary 5.11, where we show that if the assumption is replaced by the requirement that the orbit of approximates that of during a sufficiently large time, i.e. for a large number of indices , then is -frequently hypercyclic, and even sometimes frequently hypercyclic.
5.3. A criterion for -frequent hypercyclicity
Here is the version of our criterion for -frequent hypercyclicity that we will use in Section 6.
Theorem 5.14**.**
Let . Assume that there exist a dense linear subspace of with and , and a constant such that the following property holds true: for every and every , there exist and such that
- (1)
; 2. (2)
* for every .*
Then is chaotic and -frequently hypercyclic.
Démonstration.
Since is dense in , the periodic points of are dense in . So we only have to show that is -frequently hypercyclic.
As in the direct proof of Corollary 5.11 given above, we first show that given , the integer appearing in the assumption of Theorem 5.14 can be supposed to be a multiple of . Given and , there exist and such that and for every , where has to be specified. Taking small enough, we can assume (see the proof of Fact 5.12) that . Let then be such that is a multiple of , and set . Then belongs to because . Also if is small enough; and , so that for every . Since , the inequality holds true a fortiori for every .
Let now be a dense sequence of vectors of , and let be a partition of into infinite sets. We define a sequence of vectors of by setting for every . In other words, the sequence enumerates infinitely many times each point of the dense sequence , and denotes the set of all indices such that .
By induction on , we construct a sequence of vectors of and a strictly increasing sequence of integers such that
- (i)
for every ; 2. (ii)
for every ; 3. (iii)
is a multiple of the period of the vector .
We set , which is well-defined by (i), and prove that is a -frequently hypercyclic vector for . Fix . For every and every , we have by (iii)
[TABLE]
Moreover, if is a multiple of , say , we can write
[TABLE]
Hence we obtain that for any ,
[TABLE]
By (i) and (ii), it follows that for every and every , we have
[TABLE]
We deduce from this inequality that for any ,
[TABLE]
Hence is a -frequently hypercyclic vector for , and Theorem 5.14 follows. ∎
Remark 5.15**.**
The proof of Theorem 5.14 shows that admits -frequently hypercyclic vectors with the property that for every , there exists such that for every (the point being that does not depend on ). Hence there exists, for every , a sequence of integers of positive upper density such that tends to as tends to infinity. As is a periodic point for , there is no contradiction in this (see [29, Rem. 4.8]).
5.3.1. Uniform recurrence, almost periodic points and -frequent hypercyclicity
Even if this is not quite clear at first sight, the following result generalizes Theorem 5.14 (see Corollary 5.19 below).
Theorem 5.16**.**
Let . Assume that the uniformly recurrent points of are dense in , and that there exists such that the following property holds true: for any and any non-empty open subset of , one can find and an arbitrarily large integer such that for every . Then is -frequently hypercyclic.
Démonstration.
It is enough to show that for any non-empty open subset of , there exists a constant such that, for every , the open set
[TABLE]
is dense in . Indeed, if is a countable basis of open subsets of , the density of each of the sets implies that is a dense subset of consisting of -frequently hypercyclic vectors for .
So let be a non-empty open subset of . Choose a uniformly recurrent point , and such that . Since is uniformly recurrent, the set
[TABLE]
has bounded gaps. Observe that we may call this set since depends on and , and depends on . So there exist a constant and an integer such that for all intervals of of length at least . We set
[TABLE]
and we show that with this choice of , all the open sets , , are dense in . Let us fix , and a non-empty open subset of . We have to show that is non-empty. Set , which is a non-empty open subset of . By assumption, one can find and an integer such that and for every . Then belongs to and our aim is to show that belongs to . For any , we have
[TABLE]
in other words,
[TABLE]
Moreover, since the interval has length at least , we know that , i.e.
[TABLE]
It follows that
[TABLE]
and hence that belongs to . ∎
We now state and prove a few consequences of Theorem 5.16. For the first one, we need a definition: if is a bounded operator on , a point is said to be almost periodic for if, for every , the set
[TABLE]
has bounded gaps. Thus, periodic points are almost periodic and almost periodic points are uniformly recurrent. The following fact is more interesting.
Fact 5.17**.**
Let . Then any vector belonging to is almost periodic for .
Démonstration.
We first recall that since belongs to , it is uniformly recurrent for (Fact 5.6); so for any , the set has bounded gaps. Now, assuming that is non-zero, we write as , where belong to and are linearly independent. Then the restriction of to the finite-dimensional subspace is power-bounded, being diagonalizable with only unimodular eigenvalues. Since contains the -orbit of , it follows that there exists a finite constant such that, for every ,
[TABLE]
So the set with is contained in , which concludes the proof. ∎
From Theorem 5.16, we now deduce
Corollary 5.18**.**
Let . Assume that there exists such that the following property holds true: for every non-empty open subset of and every , there exists an almost periodic point such that: for any , one can find and such that and for all . Then is -frequently hypercyclic.
Démonstration.
By assumption, the uniformly recurrent points are certainly dense in . In order to check the second assumption in Theorem 5.16, let us fix a non-empty open subset of and . Let us choose a non-zero almost periodic point satisfying the assumption of Corollary 5.18, and then an integer such that any interval of of length contains a point of the set
[TABLE]
Now, choose with arbitrarily small and an arbitrarily large integer such that for every (since is non-zero, one can ensure that be arbitrarily large by taking small enough.) Having fixed and in this way, we can pick such that belongs to . Then has arbitrarily small norm, so belongs to . Moreover, since and , we have for every :
[TABLE]
The assumptions of Theorem 5.16 are thus satisfied. ∎
Here is an immediate consequence of Corollary 5.18, whose statement is a bit less convoluted. This result shows in particular that in Theorem 5.14, one can replace the assumption that the vectors of are periodic by the assumption that they are almost periodic. Moreover, it also shows that it was in fact unnecessary to assume that was a linear subspace of .
Corollary 5.19**.**
Let . Assume that there exist a dense subset of consisting of almost periodic points for , and a constant such that the following property holds true: for every and every , there exist and such that and for all . Then is -frequently hypercyclic.
Since Corollary 5.19 is so similar to Theorem 5.14, it is natural to ask whether one can provide a “constructive” proof of it resembling that of Theorem 5.14. We do so now, with the additional assumption that the above dense set is a linear subspace of .
Constructive proof of Corollary 5.19.
Define sequences , and as in the beginning of the proof of Theorem 5.14. By induction, we construct a sequence of vectors of and an increasing sequence of integers such that
- (i)
for every ; 2. (ii)
for every ; 3. (iii’)
\left\|T^{n_{j}+k}\bigl{(}\sum_{i<j}z_{i}\bigr{)}-T^{k}\bigl{(}\sum_{i<j}z_{i}\bigr{)}\right\|<2^{-j} for all .
Let . Assume that the construction has been carried out up to the step , and let us construct and . Since the vector lies in , it is almost periodic; so one can find an integer such that every interval of of length contains a point of the set
[TABLE]
By assumption on , there exists with and such that
for every ;
- -
for every .
Choose now such that belongs to the set , and let . Then (iii’) holds true by the choice of , (i) clearly holds true, and (ii) holds true as well because for every and every . This concludes the inductive step.
Let us prove that the vector is a -frequently hypercyclic vector for . This time, we write for every , and :
[TABLE]
Since belongs to , we deduce that if , then
[TABLE]
Now, since is uniformly recurrent for , the set
[TABLE]
has bounded gaps. So one can find a constant such that for all sufficiently large intervals . From this, it follows that if is large enough, then
[TABLE]
and hence that
[TABLE]
∎
Finally, here is a somewhat unexpected consequence of Theorem 5.16.
Corollary 5.20**.**
Let . Assume that the uniformly recurrent points of are dense in , and that there also exists a dense set of points such that as . Then is -frequently hypercyclic.
Démonstration.
This follows immediately from Theorem 5.16. ∎
Remark 5.21**.**
Corollary 5.20 shows in particular that if a unilateral weighted backward shift on or has a dense set of uniformly recurrent points, then it is -frequently hypercyclic. However, a much stronger result is true: if a weighted backward shift has just a single non-zero uniformly recurrent point, then it is in fact chaotic and frequently hypercyclic.
Démonstration.
Let be a weighted backward shift on or , , associated to the weight sequence , and assume that has a non-zero uniformly recurrent point . By the classical Frequent Hypercyclicity/Chaoticity Criterion (see [7] or [31]), it is enough to show that the sequence \Bigl{(}\prod_{k=1}^{n}\omega_{k}^{-1}\Bigr{)} belongs to the space . We check this in the case where , the case being similar. So we have to show that as . Since is non-zero, there exists such that . Let . Since is uniformly recurrent for , there exists a strictly increasing sequence of integers and a positive integer such that and for every . In particular, we have
[TABLE]
and hence
[TABLE]
Since belongs to , we deduce that
[TABLE]
Moreover, if is a sufficiently large integer, there exists an integer such that . Since , this implies that
[TABLE]
and this concludes the proof. ∎
Remark 5.22**.**
A natural question that comes to mind in view of Corollary 5.20 is whether any chaotic and topologically mixing operator has to be -frequently hypercyclic. We will prove in Section 6.8 that it is not the case.
5.3.2. More about -frequent hypercyclicity and c(T)
Corollary 5.20 says in essence that lots of uniformly recurrent points plus lots of orbits tending to [math] imply -frequent hypercyclicity. In the same spirit, we now prove the following result, which again may look rather surprising at first sight.
Theorem 5.23**.**
Let be hypercyclic, and assume that has a dense set of uniformly recurrent points. Then is -frequently hypercyclic if and only if .
Démonstration.
One implication is clear: if is -frequently hypercyclic, then by the very definition of .
Conversely, assume that . Then there is a comeager set such that, for every , one can find a subset of with such that as along (see [29, Prop. 4.7]). The key step of the proof lies in the next fact, where we use for convenience the following notation: if is an open subset of and is an open ball of , we write if there exists such that .
Fact 5.24**.**
Let be a uniformly recurrent point for , and let be an open neighbourhood of . Let also be an open ball with center such that , and choose an integer such that every interval of cardinality at least intersects . Then, for every , we have .
Proof of Fact 5.24.
Write , with . Since and \bigcup_{k=0}^{N-1}\bigl{(}\mathcal{N}_{T}(x_{0},B)-k\bigr{)}=\mathbb{N} by the choice of , one can find such that \overline{\textrm{dens}}\,\bigl{(}D_{x}\,\cap\,(\mathcal{N}_{T}(x_{0},B)-k)\bigr{)}\geq c(T)/N; equivalently, \overline{\textrm{dens}}\,\bigl{(}(D_{x}+k)\,\cap\,\mathcal{N}_{T}(x_{0},B)\bigr{)}\geq c(T)/N. Now, we have for all , and as along . Since , it follows that all but finitely many integers belong to , and hence that . ∎
It is now easy to conclude the proof of Theorem 5.23. Let be a countable basis of (non-empty) open sets for . Choosing for each a uniformly recurrent point and an open ball with center such that , we see that the following holds true: for each , there is an integer and a comeager set (namely, ) such that for every . Then every vector in the comeager set is a -frequently hypercyclic vector for . ∎
Remark 5.25**.**
In Theorem 5.23, one cannot replace the assumption of -frequent hypercyclicity with that of frequent hypercyclicity. Indeed, as we shall see in Theorem 6.29, there exist operators which are chaotic and -frequently hypercyclic (hence, hypercyclic with a dense set of uniformly recurrent points and such that ) but not frequently hypercyclic.
When the operator is chaotic, the proof of Theorem 5.23 gives a more precise statement, which says that if , then is “-frequently hypercyclic with estimates”.
Corollary 5.26**.**
Let be a chaotic operator on with . For any open set in , let us denote by the smallest period of all periodic vectors of belonging to . Then there is a comeager set of vectors such that
[TABLE]
Démonstration.
Note that if the point in Fact 5.24 is a periodic point of , then one can take as the period of . Then follow the proof of Theorem 5.23. ∎
From Corollary 5.26, we immediately deduce
Corollary 5.27**.**
If is chaotic and ergodic, then there is a comeager set of vectors such that for every non-empty open subset of .
Démonstration.
This is clear since for any ergodic operator . ∎
Remark 5.28**.**
All vectors satisfying the conclusion of Corollary 5.27 are -frequently hypercyclic for , but none of them is frequently hypercyclic. Indeed, if is any open ball centered at [math], then , so that . Hence, one must have for any open set disjoint from .
To conclude this section, we now proceed to prove a generalization of Corollary 5.27 to ergodic operators which are not necessarily chaotic: we require that belongs to , i.e. that the unimodular eigenvectors of span a dense subspace of .
We first need to introduce some notation. To every , we associate the rotation-invariant measure on defined as follows:
[TABLE]
Suppose now that , and that is a finite or infinite sequence of linearly independent eigenvectors associated to unimodular eigenvalues . Let us denote by the linear span of the vectors , . Any vector may be written in a unique way as
[TABLE]
where the coefficients , , are complex scalars. To any such vector , we associate the -invariant measure on defined by
[TABLE]
for every Borel subset of . Note that if the unimodular numbers are not roots of unity (and is a Hilbert space ), then is nothing but the Steinhaus measure associated to considered in the proof of Fact 4.17.
Finally, for any open set , we define
[TABLE]
As in Fact 5.24, the notation means that for some .
We may now state
Proposition 5.29**.**
Let be an ergodic operator on belonging to , and let be an infinite linearly independent sequence of unimodular eigenvectors whose linear span is dense in . Then, there exists a comeager subset of such that every vector satisfies
[TABLE]
The key step in the proof of Proposition 5.29 is Lemma 5.30 below.
Lemma 5.30**.**
Let be an ergodic operator on , and let be a finite sequence of linearly independent unimodular eigenvectors for . Let also be a vector belonging to . Given any , , there exists a -invariant measure on , ergodic for and with full support, such that
[TABLE]
Since the sequence is fixed, we will remove any reference to it in the proofs of Proposition 5.29 and Lemma 5.30; so we will write for instance instead of , and so on.
Proof of Lemma 5.30.
Let be an ergodic measure with full support for . For each , consider the probability measure defined on by setting
[TABLE]
where K_{u}:=\bigl{\{}\sum_{k=1}^{r}a_{k}(u)\mu_{k}u_{k},\;\,\mu_{k}\in\mathbb{T},\ 1\leq k\leq r\bigr{\}}. Note that is a compact subset of and contains the support of the measure . It clearly satisfies . The measure thus defined is -invariant. Indeed, we have for every Borel subset of
[TABLE]
The next step of the proof is to show that . Since
[TABLE]
and , it suffices to show that for every vector , i.e. that for every such . Write as , where and belongs to for every , and fix a vector . In order to show that belongs to , we need to show that for every and every , there exists an integer such that . Now, a result of Shkarin [47] states the following: given a hypercyclic operator on a Banach space , and a compact topological group generated by an element , the set
[TABLE]
is dense in for every vector which is hypercyclic for . We apply this result to the operator and to the subgroup of generated by . Since the -tuple belongs to , and is a hypercyclic vector for , there exists for every an integer such that and . It follows that
[TABLE]
so that . This proves our claim, and shows that for every .
Let us now estimate from below the quantities . By the definition of , we have
[TABLE]
Now, observe that for every , the set contains the ball . Indeed, every with can be written as with . It follows that
[TABLE]
Since tends to as tends to infinity, there exists such that
[TABLE]
Applying the Ergodic Decomposition Theorem (see e.g. [45, Th. 2.5]) to the measure , we obtain that there exists an ergodic measure for such that
[TABLE]
Since , the measure has full support, and this concludes the proof of Lemma 5.30. ∎
Proof of Proposition 5.29.
Let be a countable basis of (non-empty) open sets for with the following property: for any open set and any open ball such that , one can find such that . This additional property implies that for any open set , we have
[TABLE]
For each , one may choose a sequence of vectors of , a sequence of positive numbers and a sequence of positive numbers tending to [math] as tends to infinity such that
[TABLE]
By Lemma 5.30 and the pointwise ergodic theorem, we see that for each fixed pair of integers, the set of all vectors such that
[TABLE]
is dense in . In particular, the set
[TABLE]
is a dense subset of . Hence is a dense subset of too, and by the definition of , every vector satisfies
[TABLE]
Property (15) then allows us to conclude the proof of Proposition 5.29. ∎
5.4. A criterion for frequent hypercyclicity
We now move over to a criterion of the same kind as Theorem 5.14 for frequent hypercyclicity. Instead of requiring in the assumption that be small for indices less than a fraction of , we have to require that these quantities should be small for indices less than a fraction of a certain multiple of the period of . The criterion reads as follows:
Theorem 5.31**.**
Let . Assume that there exist a dense linear subspace of with and , and a constant such that the following property holds true: for every , every and every integer which is the period of some vector of , there exist and integers , such that
- (0)
* is a multiple of and of ;* 2. (1)
* for every ;* 3. (2)
* for every .*
Then is chaotic and frequently hypercyclic.
Remark 5.32**.**
The restriction that the integer above should be the period of some vector may seem rather artificial. The reason for stating Theorem 5.31 as we did is to be found in the proof of Theorem 6.9 below: we will use Theorem 5.31 as stated to prove the frequent hypercyclicity of the operators involved there without having to impose certain divisibility conditions on the numbers , .
Remark 5.33**.**
The constant involved in the statement of Theorem 5.31 cannot be greater than . Indeed, otherwise one could find a periodic vector such that more than half of the points in the orbit of are close to [math] and more than half of these points are close to some periodic orbit far away from [math]. So at least one point in the orbit of would have to be both close to [math] and far away from [math], which is impossible.
Proof of Theorem 5.31.
We first note that upon substracting some multiple of to (where and are given by the assumptions of Theorem 5.31), we can always assume that . The assumption of Theorem 5.31 is thus seen to be stronger than that of Theorem 5.14. Also, we can require that and the same argument as in the beginning of the proof of Theorem 5.14 (taking very small) shows that, given , we can always add to the assumption of Theorem 5.31 the additional hypothesis that the integer is a multiple of .
Let now be a dense sequence of vectors of contained in , and let be a partition of such that each set is infinite and has bounded gaps. We denote by the maximum size of a gap between two successive elements of . As usual, we define the vectors , , by setting for every . By induction on , we construct a sequence of vectors of and two strictly increasing sequences of integers and such that
- (i)
is a multiple of and of ; 2. (ii)
for every ; 3. (iii)
for every ; 4. (iv)
is a multiple of and ; 5. (v)
.
We set , and show that is a frequently hypercyclic vector for .
Let us fix , and write as , where is strictly increasing and for every . For each , we define a family of sets as follows:
[TABLE]
and, for ,
[TABLE]
Before doing anything with these sets, we note that
[TABLE]
Indeed, for we have because ; and the result then follows by a straightforward induction on .
Fact 5.34**.**
For every and every , , we have
[TABLE]
Proof of Fact 5.34.
For any , we have
[TABLE]
Since for every by (16), the second term in (17) is easy to control:
[TABLE]
We now have to estimate the term \displaystyle\Bigl{|}\Bigl{|}\,T^{\,n}\Bigl{(}\sum_{s=1}^{j_{m}+j}z_{s}\Bigr{)}-x_{l}\,\Bigr{|}\Bigr{|}. The index being fixed, we show by induction on that
[TABLE]
- Suppose first that belongs to , so that
[TABLE]
Then
[TABLE]
Since , we deduce from (iii) that \displaystyle\Bigl{|}\Bigl{|}\,T^{\,n}\Bigl{(}\sum_{s=1}^{j_{m}}z_{s}\Bigr{)}-x_{l}\,\Bigr{|}\Bigr{|}\leq 2^{-j_{m}}.
- Suppose now that (18) has been proved up to the index for some , and that belongs to . Write
[TABLE]
Then
[TABLE]
Thus
[TABLE]
by the induction hypothesis. Now, by (16) since belongs to , and thus by (i). Hence
[TABLE]
which concludes our induction. The inequality (18) being proved, we deduce that
[TABLE]
for every and every , . ∎
It follows from Fact 5.34 that given , there exists such that contains the set
[TABLE]
So in order to show that is a frequently hypercyclic vector for , it suffices to prove the following fact.
Fact 5.35**.**
The set has positive lower density.
Proof of Fact 5.35.
We start with a series of elementary remarks on the structure of the sets .
- (a)
and for , so that the sets are contained in successive (disjoint) subintervals of . 2. (b)
Inside each set , , the translates are pairwise disjoint. 3. (c)
If we set , then . Since , the sets , , are thus contained in successive subintervals of ; 4. (d)
There is no redundancy in the definition of : if
[TABLE]
then and . Indeed, we have , so that , and hence because . 5. (e)
It follows from (v) that
[TABLE] 6. (f)
By (a) and (v), we have for every :
[TABLE]
Iterating this inequality yields that
[TABLE]
Since , this eventually gives
[TABLE]
In order to prove that has positive lower density, we will show the existence of some positive number such that
[TABLE]
This will be enough to ensure that has positive lower density. Indeed, if we enumerate the set as where is strictly increasing, this inequality can be rewritten as for every , i.e. ; and this is easily seen to imply that has positive lower density.
We fix and , and consider separately two cases.
- Suppose that , and write
[TABLE]
Note that the set of (c) is contained in because , and is disjoint from . So we have in particular
[TABLE]
Moreover, the set contains all integers of the form , where and (since all of them are in and the largest is not greater than ). Hence,
[TABLE]
By (f), it follows that
[TABLE]
Since , we conclude that
[TABLE]
- Suppose now that for some , so that
[TABLE]
Then by (16). In particular,
[TABLE]
Moreover, since and the translates , are pairwise disjoint (and contained in ), we also have
[TABLE]
By (f), it follows that
[TABLE]
where we have used the inequality .
We have thus proved that
[TABLE]
which concludes the proof of Fact 5.35. ∎
The proof of Theorem 5.31 is now complete. ∎
5.4.1. Link with the Operator Specification Property
We conclude this section by a result which shows that many operators which are known to be frequently hypercyclic satisfy the assumption of Theorem 5.31. This is the case for all operators which satisfy the Frequent Hypercyclicity Criterion and, more generally, for operators with the so-called Operator Specification Property.
This last property, which has been recently introduced and studied by Bartoll, Martínez-Giménez, and Peris ([1], [2]), is the linear version of the classical Specification Property for compact dynamical systems introduced by Bowen in [15]. The definition reads as follows: an operator has the Operator Specification Property (OSP) if there exists an increasing sequence of -invariant subsets of with , the union of which is dense in , such that for each , the restriction of to has the Specification Property in the sense of [15], which means that the following holds true:
for every , there exists an integer such that for every finite family of points of , and any integers with for every , there is a point such that for every ,
[TABLE]
Stated in an informal way, means that arbitrary large pieces of the orbits of finitely many points of can be approximated by the orbits of a single periodic point of , provided that the gaps between the different sets of indices where we require this approximation are sufficiently large.
It is proved in [2] that operators with the OSP are chaotic, topologically mixing, and frequently hypercyclic. Moreover, any operator satisfying the general version of the Frequent Hypercyclicity Criterion given in [13] has the OSP. We prove in Theorem 5.36 below that operators with the OSP satisfy the assumption of Theorem 5.31. This was pointed out to us by Alfred Peris, who kindly allowed us to reproduce his proof here. This improves on a previous observation (proved in a preliminary version of this paper) according to which operators satisfying the Frequent Hypercyclicity Criterion also satisfy the assumptions of Theorem 5.31.
Theorem 5.36**.**
Let be an operator with the OSP. There exists a dense subspace of with and , such that, for every , every , every , and every integer , there exist and integers , such that properties (0), (1), and (2) of Theorem 5.31 hold true.
Démonstration.
Let be an increasing sequence of -invariant subsets of with , the union of which is dense in , such that above holds true for every . We define X_{0}:=\textrm{span}\bigl{[}\bigcup_{m\geq 1}K_{m}\cap\textrm{Per}(T)\bigr{]}, which is clearly -invariant and dense in .
Let us fix , and . The vector can be written as , where is a scalar and is a vector belonging to for every . By [2, Prop. 10], the map induced by on the set has the Specification Property. So we can assume without loss of generality that belongs to for some . Let now be such that property above holds true, and let be a multiple of so large that . We then define to be the integer part of , , , and , where is any multiple of such that . We also set , , , and . As , the Specification Property on implies that there exists a point such that
[TABLE]
since ; and is a multiple of the period of . Since , this shows that assumptions (0), (1), and (2) of Theorem 5.31 are satisfied. ∎
Remark 5.37**.**
The converse of Theorem 5.36 is not true; that is, operators satisfying the assumptions of Theorem 5.31 need not have the OSP. Indeed, we will construct in Section 6 (more precisely, in Example 6.23) some operators which satisfy the assumptions of Theorem 5.31 and yet are not topologically mixing.
To conclude this section, we essentially show that the OSP implies ergodicity. This is coherent with what happens in the non-linear setting: it is proved in [50] that compact dynamical systems with the Specification Property are ergodic. They actually enjoy a much stronger property: given such a system , the ergodic measures with full support for are comeager in the set of -invariant probability measures endowed with its natural Polish topology. On the other hand, we do not know whether operators with the OSP are mixing (i.e. admit mixing measures with full support), whereas it is known that the Frequent Hypercyclicity Criterion does imply mixing, and in fact mixing in the Gaussian sense (see [41] and [8]).
Proposition 5.38**.**
If satisfies the OSP, and if the sequence of -invariant subsets of appearing in the definition of the OSP consists of compact sets, then is ergodic.
The assumption of Proposition 5.38 that the sets , , be compact seems to be no real restriction. As mentioned in [2], all known examples of operators with the OSP do satisfy the OSP with respect to a sequence of -invariant compact subsets of .
Démonstration.
Let us denote by the space of all Borel probability measures on endowed with its natural Polish topology (a sequence of elements of converges to if and only if tends to as tends to infinity for every bounded continuous function from into ), and by the set of all -invariant measures. Then is a Polish space, being a closed subset of . We denote by the set of all ergodic measures for . For any family of measures , we denote by the family of all measures with full support. Finally, for any subset of and any Borel subset of , we set .
Let be an increasing sequence of compact -invariant subsets of with such that has the Specification Property for every . By a result of [50], we know that for each , the set of all measures with support equal to is dense in . Let us now denote by the closure of in .
Fact 5.39**.**
The set is a dense subset of .
Proof of Fact 5.39.
It is known that is a subset of (see [42] for a detailed proof), so that is in . Moreover, since is dense in for each by [50], it is clear that is dense in . ∎
Fact 5.40**.**
The set is a dense subset of .
Proof of Fact 5.40.
Let be a countable basis of non-empty open subsets of . Set, for each , . Then each set is open in , and moreover
[TABLE]
So we just have to show that each set is dense in . Let us fix , and let be a non-empty open subset of . Since the sequence is increasing, the definition of implies that is non-empty for all sufficiently large. Moreover, since is dense in and is increasing, is non-empty too for all sufficiently large. So there exists such that and are both non-empty. Since is dense in and is open in , it follows that there exists a measure with support equal to such that belongs to and . Hence, we have shown that is non-empty. ∎
The two facts above combined with the Baire Category Theorem applied in imply that is non-empty. In particular is non-empty, i.e. is ergodic. ∎
Remark 5.41**.**
We will see in Section 6 that the assumptions of Theorem 5.31 (which are sufficient for frequent hypercyclicity) do not imply ergodicity. So Proposition 5.38 makes the difference between the OSP and our criterion for frequent hypercyclicity all the more tangible.
6. Special examples of hypercyclic operators
In this section, we introduce some particular classes of operators, which are defined on any space , . We do not restrict ourselves to the Hilbertian case , because the general case adds no extra complication. It is within these classes that we will exhibit operators which are chaotic and frequently hypercyclic but not ergodic, operators which are chaotic and -frequently hypercyclic but not frequently hypercyclic and operators which are chaotic and topologically mixing but not -frequently hypercyclic.
A word of caution: for technical reasons, we have decided that starts at [math]; that is, . Accordingly, the canonical basis of will be denoted by . Also, we denote by the linear span of the vectors , , i.e. the subspace of consisting of all finitely supported vectors.
6.1. Operators of C-type: basic facts
In view of our criteria for -frequent hypercyclicity and frequent hypercyclicity relying on the existence of periodic points, we would like to find a rich family of operators for which we can easily find a large supply of periodic points. For example, we could consider operators defined by
[TABLE]
where is a weight sequence and is a strictly increasing sequence of integers with . Indeed, with this definition it is clear that (whatever the choice of and ) every basis vector and hence every vector is periodic for .
However, none of these operators is hypercyclic since they are direct sums of finite-dimensional operators, and there exist no hypercyclic operators in finite dimension. It is thus necessary to perturb the operators in order to obtain an interesting family of hypercyclic operators in which finite sequences are still periodic. For reasons that will be explained below, the operators in this family will be called operators of C-type. Any operator of C-type will be associated to four parameters , , , and , where
is a sequence of non-zero complex numbers such that ; 2. -
is a sequence of complex numbers which is both bounded and bounded below, i.e. ; 3. -
is a map from into itself, such that , for every , and the set is infinite for every ; 4. -
is a strictly increasing sequence of positive integers such that and is a multiple of for every .
Definition 6.1**.**
The operator of C-type on associated to the data , , , and given as above is defined by
[TABLE]
Note that by convention, an empty product is declared to be be equal to ; that is, when (which can happen only for ).
Without any additional assumption on the parameters, these formulas define a linear map on only. The first issue is of course the boundedness of .
Fact 6.2**.**
The operator is well-defined and bounded on as soon as that the following condition holds true:
[TABLE]
Démonstration.
This is rather clear. Indeed, it appears that can be written as
[TABLE]
where is the operator defined by
[TABLE]
which is clearly bounded (and even compact) because , and, for each , is a finite-dimensional cyclic operator acting on . Condition implies that is finite, and is thus bounded. ∎
From now on, we will always assume that Condition is satisfied. Also, when no confusion arises, we will write simply instead of .
Remark 6.3**.**
We call such operators operators of C-type for two different reasons. On the one hand, “C” stands for “cyclic”: as we have just explained, each operator is a compact perturbation of an infinite direct sum of cyclic finite-dimensional operators , where is defined on . On the other hand, we will see in a moment that, as a consequence of their particular structure, the operators happen to be chaotic under a very mild restriction on the parameters; so, “C” stands for “chaotic” as well.
The following identity will be used repeatedly: if is an operator of C-type on , then
[TABLE]
This allows to show that operators of C-type always have plenty of periodic points:
Fact 6.4**.**
If is an operator of C-type on , then every basis vector is periodic for ; more precisely,
[TABLE]
Consequently, every vector is periodic for , and hence has a dense set of periodic points.
Démonstration.
Since is a non-zero multiple of for every , it suffices to prove that for every . We prove this by induction on .
– If , then by definition of , and thus .
– Let , and assume that the result has been proved for all . We know that is a multiple of and since , it follows by (21) and the induction hypothesis that
[TABLE]
∎
Using the above fact and Corollary 5.13, we can now obtain the following sufficient condition for an operator of C-type to be chaotic.
Proposition 6.5**.**
Suppose that
[TABLE]
Then the operator of C-type on is chaotic.
Démonstration.
We apply Corollary 5.13 with . Fix and . We are looking for a vector and an integer such that and . Let be such that belongs to . By the assumption of Proposition 6.5, there exists such that
[TABLE]
The vector
[TABLE]
is periodic by Fact 6.4, and satisfies by (22). Moreover, since , we have by (21):
[TABLE]
By definition of , this implies that
[TABLE]
and by (22), it follows that . The assumptions of Corollary 5.13 are thus satisfied, and is chaotic. ∎
Recall that every chaotic operator is topologically weakly mixing [7, Ch. 4] and reiteratively hypercyclic [12]. In the following subsections, we will be interested in frequent hypercyclicity, -frequent hypercyclicity and topological mixing for operators of C-type.
6.2. Operators of C*+*-type: how to be FHC or UFHC
For the construction of our counterexamples, we will work with operators of C-type for which the data , , and have a special structure. For every integer , we require that:
for every , so that ; 2. -
the blocks , , all have the same size, which we denote by :
[TABLE] 3. -
the sequence is constant on the interval : there exists such that
[TABLE] 4. -
the sequences of weights are independent of : there exists a sequence such that
[TABLE]
If these conditions are met, we say that is an operator of C+-type.
Remark 6.6**.**
By definition, the map is the same for all operators of C*+*-type, so it is no longer a “parameter”. However, we will continue using the notation .
Our first result concerning these operators gives a sufficient set of conditions for a C*+*-type operator to be -frequently hypercyclic. This will be deduced from Theorem 5.14.
Theorem 6.7**.**
Let be an operator of C+-type on . Suppose that there exists a constant such that the following property holds true : for every and every integer , there exist two integers and such that
[TABLE]
Then is chaotic and -frequently hypercyclic.
Démonstration.
The following fact will be useful.
Fact 6.8**.**
Let . For any and , we have
[TABLE]
Proof of Fact 6.8.
Since , we have . So the formula follows directly from the definition of . ∎
We are going to show that the assumption of Theorem 5.14 is satisfied with . So let us fix and . We choose such that may be written as
[TABLE]
Let be a very large number, to be specified later on in the proof. By assumption, there exist and such that
[TABLE]
and
[TABLE]
Note that since the sequences and are bounded, it follows from (23) that the integer can be chosen as large as we please, provided that is large enough. So we may assume from the beginning that . We will also assume that , which will be useful below.
We set
[TABLE]
Our aim is to prove that if has been suitably chosen, then
[TABLE]
Theorem 5.14 will then conclude the proof.
The first of these two claims is the easiest one to prove. Indeed, since the weight sequence is bounded and bounded from below, it follows from (23) that
[TABLE]
where is an absolute constant. So if is large enough.
Let us now estimate the norm of the vector for every . Note that if and , then since and . Applying Fact 6.8 with , we get
[TABLE]
and hence
[TABLE]
because T^{j-b_{l}}e_{b_{l}}=\Bigl{(}\,\,\prod_{i=1}^{j-b_{l}}w_{b_{l}+i}\Bigr{)}\,e_{j} and T^{j-b_{l}}e_{b_{2^{k-1}+l}}=\Bigl{(}\prod_{i=1}^{j-b_{l}}w^{(k)}_{i}\Bigr{)}\,e_{b_{2^{k-1}+l}+j-b_{l}}.
Moreover, if , then
[TABLE]
because . So we get
[TABLE]
By definition of , it follows that for any , we have
[TABLE]
By (24) and since the weight sequence is bounded and bounded below, this implies that
[TABLE]
where (as above) is an absolute constant. So we have for every if is large enough.
The assumptions of Theorem 5.14 are thus satisfied, and this concludes the proof of Theorem 6.7. ∎
Using Theorem 5.31 instead of Theorem 5.14, we can obtain sufficient conditions of the same kind as above for operators of C*+*-type to be frequently hypercyclic.
Theorem 6.9**.**
Let be an operator of C+-type on . Suppose that there exists a constant such that the following property holds true: for every and every , there exist two integers and such that
[TABLE]
Then is chaotic and frequently hypercyclic.
Démonstration.
The proof of Theorem 6.9 is so similar to that of Theorem 6.7 that we only sketch it very briefly. The role of the integer in the assumption of Theorem 5.31 is played by the integer , which is a period of . If is chosen sufficiently large at the beginning of the proof, can be supposed to be a multiple of the period of any fixed vector : indeed, such a vector always has a period of the form for some integer , and divides as soon as since , and ∎
Now that we have obtained sufficient conditions for the frequent and -frequent hypercyclicity of operators of C*+*-type, we need to find necessary conditions as well. This we do in the next subsection.
6.3. Operators of C-type: how not to be FHC or UFHC
Since -frequent hypercyclicity and frequent hypercyclicity are strong notions of hypercyclicity, one might think that it is easier to prove that an operator does not have one of these properties, than to prove that an operator has it. However, instead of exhibiting a single -frequent or frequent hypercyclic vector, we now need to prove that no vector of whatsoever can be -frequent or frequent hypercyclic; and put in this way, this no longer looks that easy.
In this subsection, we are going to single out some conditions ensuring that an operator of C-type is not -frequently hypercyclic or not frequently hypercyclic. As suggested in the few lines above, the arguments will be rather more technical than in the previous subsection. However, we can give the basic idea immediately: if for every hypercyclic vector for , we are able to find some such that the set has upper density (resp. lower density) equal to [math], then will not be -frequently hypercyclic (resp. frequently hypercyclic).
6.3.1. The main criterion, in abstract form
The following notation will be used throughout this subsection: given an operator of C-type , we denote for each by the canonical projection of onto : if , then
[TABLE]
The following theorem provides sufficient conditions for an operator of C-type to be non--frequently hypercyclic or non-frequently hypercyclic. These conditions are stated in terms of the projections .
Theorem 6.10**.**
Let be an operator of C-type on . Suppose that for every hypercyclic vector for , there exist
a positive constant , 2. -
a non-increasing sequence of positive real numbers with , 3. -
a sequence of non-negative real numbers, 4. -
a non-decreasing sequence of integers tending to infinity,
such that the following conditions are satisfied:
- (1)
* for every ;* 2. (2)
* for every and every ;* 3. (3)
* for every and every ;* 4. (C)
\liminf\limits_{l\to\infty}\ \ \inf\limits_{k\geq N_{l}}\dfrac{\ \#\bigl{\{}0\leq j\leq k\,;\,\|P_{l}\,T^{\,j}P_{l}\,x\|\geq 2CX_{l}\bigr{\}}}{k+1}=1.**
Then is not -frequently hypercyclic.
If Condition (C) is replaced by
- (C’)
\liminf\limits_{l\to\infty}\ \ \inf\limits_{k\geq N_{\min(\varphi^{-1}(l))}}\dfrac{\ \#\bigl{\{}0\leq j\leq k\,;\,\|P_{l}\,T^{\,j}P_{l}\,x\|\geq 2CX_{l}\bigr{\}}}{k+1}=1,
then is not frequently hypercyclic.
The usefulness of this result lies in the fact that the lower bounds for the densities are given in terms of norms which are easily computable. Moreover, since is periodic, we can determine the cardinality of the sets by examining only a fixed finite number of iterates (independent of ).
Theorem 6.10 will follow very easily from the next lemma, which provides, under conditions (1), (2), and (3) above on the projections , lower bounds for the upper and lower density of some sets , where is any non-zero vector of and is a positive number depending on . Here, of course, denotes the complement of the the ball in .
Lemma 6.11**.**
Let be an operator of C-type on . Fix and suppose that there exist
a positive constant , 2. -
a non-increasing sequence of positive real numbers with , 3. -
a sequence of non-negative real numbers, 4. -
a non-decreasing sequence of integers tending to infinity,
such that the following conditions are satisfied:
- (1)
* for every ;* 2. (2)
* for every and every ;* 3. (3)
* for every and every ;* 4. (4)
* for every .*
Then there exists such that
[TABLE]
and
[TABLE]
Let us show how this lemma implies Theorem 6.10.
Proof of Theorem 6.10.
For every and every , the vector is a periodic vector for . Hence, if is hypercyclic for then
[TABLE]
Indeed, otherwise we would have
[TABLE]
a contradiction with the hypercyclicity of .
By (27), condition (4) of Lemma 6.11 is satisfied for any choice of the sequence . It follows that if condition (C) in Theorem 6.10 is satisfied, then, for any , one can find such that \overline{\rm dens}\,\bigl{(}\mathcal{N}_{T}(x,B(0,\varepsilon)\bigr{)}=0; whereas if (C’) is satisfied, then, for any , one can find such that \underline{\rm dens}\,\bigl{(}\mathcal{N}_{T}(x,B(0,\varepsilon)\bigr{)}=0. This concludes the proof. ∎
Before giving the proof of Lemma 6.11, let us explain the general idea. The starting point is given by the inequalities
[TABLE]
In other words, we get a lower bound for as soon as we have a lower bound for some quantity and an upper bound for , . The role of the lower bound for will be played by , which will be bigger than . Assumption (3) will provide us with an upper bound for when is smaller than . However, condition (4) tells us that will be large for some and some . Assumption (2) will then be used to deduce that , which will allow us to deduce that and to repeat the above arguments for , and so on.
Proof of Lemma 6.11.
If is a non-zero periodic vector for , one can obviously choose such that
[TABLE]
and the conclusion of Lemma 6.11 is satisfied. So we henceforward suppose that is not a periodic vector for .
We first note that there exists an integer such that
[TABLE]
Indeed, since it is impossible to have for every . Since is not periodic, is non-zero. Assumption (1) of Lemma 6.11 implies thus that is non-zero and we set .
We now construct, by induction on , a strictly increasing sequence of integers such that if we set
[TABLE]
Observe that this does make sense: the set involved in the definition of is non-empty by assumption (4).
Suppose that the integers have already been constructed. Then there exists an integer with the property that
[TABLE]
Indeed, if we had for every , this would imply that , violating the definition of .
By assumption (2), we have
[TABLE]
In order to show that , we observe that for every ,
[TABLE]
by assumption (3) and the definition of . Since the sequence is decreasing, this yields that . Also, the induction hypothesis implies that , so that for every . Inequality (28) thus implies that and this finishes the induction. Note that the sequence tends to infinity as tends to infinity.
For any integer , let us denote by be the smallest integer such that belongs to . Then tends to infinity as tends to infinity. Since
[TABLE]
for every and , we have
[TABLE]
It follows from this inequality and the fact that for every that
[TABLE]
Hence, since , we have
[TABLE]
Now , so that belongs to the interval for each . Thus
[TABLE]
and
[TABLE]
since as . This proves the first part of Lemma 6.11.
As for the second part, we proceed in the same way, starting from the inequality
[TABLE]
Our first observation is that for every . This follows, in a slightly roundabout way, from the fact that is non-zero. Indeed, the definition of implies that there exists an integer such that . Then belongs to , and since , it follows that .
We now claim that there exists a strictly increasing sequence of integers such that for every . Recall that for all . Thus there exists such that . As belongs to the interval , we have , and thus . Extracting if necessary in order to make the sequence strictly increasing, we obtain the sequence we are looking for.
We now have
[TABLE]
This concludes the proof of Lemma 6.11. ∎
6.3.2. How to check the assumptions
The assumptions of Theorem 6.10 are partly of an “abstract” nature, since they involve the projections and not explicitly the parameters , , , and defining . We now provide concrete conditions on , , , and which imply that the operator of C-type satisfies the assumptions of Theorem 6.10.
As a rule, conditions (1), (2), and (3) will be obtained by requiring that the products of weights
[TABLE]
decrease sufficiently rapidly and by setting, given a hypercyclic vector for ,
[TABLE]
More precisely, we have the following fact, which generalizes Claims 5 and 6 in [39].
Fact 6.12**.**
Let be an operator of C-type on , and let be a sequence of positive numbers with . Assume that
[TABLE]
Then, for any , we have for every and every ,
- (1)
\quad{\displaystyle\sup_{j\geq 0}\ \|P_{n}T^{\,j}P_{l}\,x\|\leq C_{l}\,(b_{l+1}-b_{l})^{\frac{p-1}{p}}\ \Bigl{\|}\sum_{k=b_{l}}^{b_{l+1}-1}\Bigl{(}\prod_{s=k+1}^{b_{l+1}-1}w_{s}\Bigr{)}\,x_{k}e_{k}\Bigr{\|}}
and 2. (2)
\quad{\displaystyle\sup_{j\leq N}\|P_{n}T^{\,j}P_{l}\,x\|\leq C_{l}\,(b_{l+1}-b_{l})^{\frac{p-1}{p}}\Bigl{(}\ \sup_{b_{l+1}-N\leq k<b_{l+1}}\prod_{s=k+1}^{b_{l+1}-1}|w_{s}|\Bigr{)}\|P_{l}\,x\|}
for every .
Démonstration.
Fix and . We first remark that if for all , the definition of implies that for every (this argument was already used at the end of the proof of Lemma 6.11). So the required inequalities are obvious in this case.
Suppose now that for some integer . In this case, we claim that
[TABLE]
for every . This inequality will allow us to run an induction procedure in order to estimate the quantity .
In order to prove (29), we fix and we consider separately several cases.
-
Suppose first that . Then is a scalar multiple of , and .
-
Now, suppose that . Then
[TABLE]
- Finally, suppose that . In this case, we write
[TABLE]
By assumption, we have ; that is,
[TABLE]
Since , it follows that
[TABLE]
and hence that .
We have thus proved the required inequality for every . Since , it holds true in fact for every , and this proves (29).
Let now be the minimum of the integers such that . We start from the straightforward estimate
[TABLE]
and apply (29). This gives
[TABLE]
where denotes the norm of a vector .
By induction, we obtain
[TABLE]
Applying the assumption of Fact 6.12, it follows that
[TABLE]
Now, the definition of and the fact that show that
[TABLE]
Since , it follows (by the assumption of Fact 6.12), that
[TABLE]
because all the constants are supposed to belong to and . By Hölder’s inequality, we conclude that
[TABLE]
which proves assertion (1) of Fact 6.12.
The proof of assertion (2) is now straightforward. Indeed, we have for every and every :
[TABLE]
Since the first term belongs to the linear span of the vectors , , it follows that
[TABLE]
and so, by the already proved assertion (1) of Fact 6.12,
[TABLE]
This concludes the proof of Fact 6.12. ∎
The next fact provides conditions on the parameters of an operator of C-type ensuring that assumption (C) or (C’) of Theorem 6.10 is satisfied. This generalizes Claim 7 of [39].
Fact 6.13**.**
Let be an operator of C-type on , and let . Fix , and define
[TABLE]
Suppose that there exist two integers such that
[TABLE]
Then we have for every
[TABLE]
Démonstration.
Fix and , and set . Then
[TABLE]
(the plus or minus sign appearing in these equalities depends on the parity of the unique integer such that belongs to the interval ).
If we suppose that belongs to , we have
[TABLE]
and thus there exists such that
[TABLE]
Setting, for every ,
[TABLE]
we obtain that
[TABLE]
satisfies
[TABLE]
We now consider separately two cases:
- Suppose first that \displaystyle\Bigl{\|}\sum_{k\,\in\,I_{j}}x_{k}\,\Bigl{(}\prod_{s=k+1}^{b_{l+1}-1}w_{s}\Bigr{)}\,e_{k}\,\Bigr{\|}<X_{l}/2 for every . Then we have for every , and thus
[TABLE]
- Now, suppose that there exists such that
[TABLE]
Then we have for every integer such that is empty:
[TABLE]
It follows that, for every ,
[TABLE]
Now, we remark that if we set for every , we have
[TABLE]
and for every . This particular structure of the sets implies that
[TABLE]
Plugging this into (30) yields that
[TABLE]
which is the inequality we were looking for. ∎
Facts 6.12 and 6.13 will be repeatedly used in Subsections 6.5, 6.7 and 6.8 in order to exhibit the desired counterexamples.
6.4. Operators of C-type: mixing or not mixing
In this subsection, we give some conditions ensuring that an operator of C-type is or is not topologically mixing. This will allow us to show in particular in Section 6.5.2 that the assumptions of our criterion for frequent hypercyclicity (Theorem 5.31) do not imply the Operator Specification Property.
Proposition 6.14**.**
Let be an operator of C-type on . Suppose that for every and every , the set
[TABLE]
where . Then is topologically mixing.
Démonstration.
Since has a dense set of periodic points, Corollary 5.4 implies that it suffices to show that for every and every , there exists an integer such that for every , there exists a vector such that and .
Fix thus and . Let also be such that belongs to , and let be a sufficiently small positive number, to be fixed later on in the proof. By assumption, the set is cofinite. So it suffices to show that if is small enough, then, for every , one can find a vector such that
[TABLE]
Fix , so that belongs to for some integer of and
[TABLE]
We define a vector by setting
[TABLE]
Then
[TABLE]
so that for every if is small enough. Moreover, since we also have
[TABLE]
and hence (since belongs to )
[TABLE]
Since the sequence of weights is bounded by a positive constant ,
[TABLE]
and the quantity on the right hand side does not depend on . So we also have for every provided is small enough. ∎
For operators of C*+*-type, the statement of Proposition 6.14 can be slightly simplified.
Corollary 6.15**.**
Let be an operator of C+-type on . Suppose that for every , the set
[TABLE]
where . Then is topologically mixing.
Démonstration.
This follows immediately from the special structure of C*+*-type operators and the definition of the map in this case. ∎
We now use Fact 6.12 in order to formulate sufficient conditions for an operator of C-type to be not topologically mixing.
Proposition 6.16**.**
Let be a C-type operator on . Let be a sequence of positive numbers with for every . Assume that
[TABLE]
and that there exists a positive constant such that for infinitely many integers , we have
[TABLE]
Then is not topologically mixing.
Démonstration.
Recall that we denote for each by the canonical projection of onto the finite-dimensional space . It is enough to show that if satisfies the assumption (31), then
[TABLE]
Indeed, this will imply that does not belong to \mathcal{N}_{T}\bigl{(}B(0,\frac{1}{1+K}),B(3e_{0},1)\bigr{)} for every such , and hence that is not topologically mixing.
Let us fix a vector x\in B\bigl{(}0,\frac{1}{1+K}\bigr{)} and an integer satisfying (31). Recalling that for every , we have
[TABLE]
Moreover, if , it follows from Fact 6.12 that
[TABLE]
So we get that
[TABLE]
which proves Proposition 6.16. ∎
After these technical preliminaries, we are now going to consider special classes of operators of C-type, for which the general conditions for frequent hypercyclicity, -frequent hypercyclicity and topological mixing obtained previously become rather transparent. This will allow us to derive easily the promised counterexamples.
6.5. Operators of C*+,1*-type: FHC does not imply ergodic
In this subsection, we restrict ourselves to operators of C*+*-type for which the parameters , , and satisfy the following conditions: for every ,
[TABLE]
where and are two strictly increasing sequences of integers with for every . We call such operators operators of C+,1-type.
If, in order to simplify matters, we assume that for every , then the key parameter of our counterexamples will be the quantity , i.e. the proportion of weights equal to in each block . For instance, if we consider the vector , we remark that is close to , and that the orbit of follows the orbit of during an interval of time proportional to . Since the period of is equal to , it seems plausible in view of Theorem 5.31 that will be frequently hypercyclic as soon as . On the other hand, we will show that if is too small, cannot be ergodic.
6.5.1. How to be FHC or UFHC
Our first result gives a readable sufficient condition for an operator of C*+,1*-type to be chaotic and frequently hypercyclic.
Theorem 6.17**.**
Let be an operator of C+,1-type on .
- (1)
If , then is chaotic. 2. (2)
If , then is chaotic and frequently hypercyclic.
Démonstration.
The first statement is a direct consequence of Proposition 6.5: for every , the expressions of and combined with the fact that for every yield that
[TABLE]
As to the second statement, it is a consequence of Theorem 6.9. Let us fix a real number with
[TABLE]
Fix also and . By our assumption and the fact that tends to infinity as tends to infinity, there exists an integer such that
[TABLE]
We apply Theorem 6.9 with . We have
[TABLE]
which gives condition (25) of Theorem 6.9. As for condition (26), we have for every :
[TABLE]
It thus follows from Theorem 6.9 that is frequently hypercyclic. ∎
The next result gives, under some additional assumptions, necessary and sufficient conditions for an operator of C*+,1*-type to be -frequently hypercyclic or frequently hypercyclic (which turn out to be same under the additional assumptions).
Theorem 6.18**.**
Let be an operator of C+,1-type on . We additionally assume that the sequence defined by \gamma_{k}:=2^{\,\delta^{(k-1)}-\tau^{(k)}}\bigl{(}\Delta^{(k)}\bigr{)}^{1-\frac{1}{p}} for every is a non-increasing sequence, and that the following two conditions hold true:
[TABLE]
Then the following assertions are equivalent:
- (1)
* is -frequently hypercyclic;* 2. (2)
* is frequently hypercyclic;* 3. (3)
.
Démonstration.
Since , Condition (3) is equivalent to . It follows from this observation and Theorem 6.17 that (3) implies (2), which of course implies (1). It remains to prove that if , is not -frequently hypercyclic. Given a hypercyclic vector for , we need to show that cannot be a -frequently hypercyclic vector for , and for this it suffices to find , , , and satisfying assumptions (1), (2), (3) and (C) of Theorem 6.10. This is done thanks to Facts 6.12 and 6.13. We first fix the sequence by setting
[TABLE]
Since the sequence is non-increasing, the sequence is non-increasing. Also
[TABLE]
by assumption. We have for every and every
[TABLE]
If we set for every , Fact 6.12 implies that for every , every and every ,
[TABLE]
and, for every ,
[TABLE]
We now set for every and every . Remembering that if , we obtain that
[TABLE]
Finally, we set for every and every
[TABLE]
Obviously, , and we have proved that for every ,
[TABLE]
It remains to check that condition (C) of Theorem 6.10 holds true, and for this we use Fact 6.13.
We have for every , and The assumptions of Fact 6.13 are thus satisfied for and if , and we get
[TABLE]
for every , every , and every . Therefore,
[TABLE]
for every and every , and it follows that
[TABLE]
since by assumption. Theorem 6.10 eventually yields that is not -frequently hypercyclic, and this concludes the proof. ∎
Remark 6.19**.**
Theorem 6.18 implies the main result of [39], i.e. that there exist chaotic operators on which are not -frequently hypercyclic.
6.5.2. A word about the OSP
In this short subsection, we show that there exist C*+,1*-type operators on which satisfy the assumptions of the criterion for frequent hypercyclicity stated in Theorem 5.31, and yet do not have the Operator Specification Property (recall that we have proved in Section 5.4.1 that operators with the OSP do satisfy the assumptions of Theorem 5.31). The following corollary is a simple consequence of Proposition 6.16 applied to C*+,1*-type operators.
Corollary 6.20**.**
Let be an operator of C+,1-type on and let \gamma_{k}=2^{\,\delta^{(k-1)}-\tau^{(k)}}\bigl{(}\Delta^{(k)}\bigr{)}^{1-\frac{1}{p}} for every . Assume that for every and that there exists a positive constant such that, for every ,
[TABLE]
Then is not topologically mixing.
Démonstration.
Let be an integer, and let be the unique integer such that . Then and , and setting
[TABLE]
we have Moreover, if is an integer of the form for some integer , then , and moreover for every . It follows that
[TABLE]
is less than
[TABLE]
Hence Proposition 6.16 applies, and is not topologically mixing. ∎
Since the condition appearing in Corollary 6.20 is compatible with those of Theorem 6.18 above (see Example 6.23), and since operators with the OSP are topologically mixing, we immediately deduce:
Corollary 6.21**.**
There exist operators on which satisfy the assumptions of Theorem 6.18 and yet do not have the Operator Specification Property.
Since the assumptions of Theorem 6.18 imply those of Theorem 5.31, it follows that there exist operators on which satisfy the criterion for frequent hypercyclicity stated in Theorem 5.31 without having the OSP. In particular, this criterion for frequent hypercyclicity is strictly stronger than the “classical” one.
6.5.3. FHC but not ergodic
We are now ready to give examples of operators of C*+,1*-type which are frequently hypercyclic but not ergodic.
Theorem 6.22**.**
Let be an operator of C+,1-type on , and set, for every , \gamma_{k}:=2^{\,\delta^{(k-1)}-\tau^{(k)}}\bigl{(}\Delta^{(k)}\bigr{)}^{1-\frac{1}{p}}. Suppose that the sequence is non-increasing, and that the following three conditions hold true:
[TABLE]
Then is frequently hypercyclic but , so that is not ergodic.
Démonstration.
The frequent hypercyclicity of follows from Theorem 6.18. Let us show that . Recall that is characterized by the following property (see [29, Rem. 4.6]): for quasi-all hypercyclic vectors for in the Baire category sense, we have
[TABLE]
Let us fix a hypercyclic vector for such that (32) holds true. By Lemma 6.11 combined with the end of the proof of Theorem 6.18, we know that there exists an such that
[TABLE]
(recall that we proved above that under the assumption (3) of Theorem 6.18, the assumptions of Theorem 6.10 are satisfied with , so that ). Using an inequality obtained at the end of the proof of Theorem 6.18, we get
[TABLE]
It follows that
[TABLE]
Our assumption thus implies that
[TABLE]
which proves that is not ergodic. ∎
Example 6.23**.**
Let be a positive integer, and consider the operator of C*+,1*-type on associated to the parameters
[TABLE]
If is sufficiently large, is frequently hypercyclic but not ergodic. Besides, it is also not topologically mixing.
Démonstration.
With this choice of parameters, and assuming that , we have
[TABLE]
for every . So it is not hard to check that if is sufficiently large, the sequence is non-increasing and satisfies . The other assumptions of Theorem 6.22 are clearly satisfied, and hence is frequently hypercyclic but not ergodic.
In order to show that is also not topologically mixing if is large enough, we use Proposition 6.16. If is any integer of the form , where is any integer, then
[TABLE]
If is sufficiently large, then ; so we deduce from Proposition 6.16 (by considering and for every ) that is not topologically mixing. Notice that also satisfied the assumptions of Corollary 6.20. ∎
Since the conditions of Theorem 6.18 make no difference between -frequent and frequent hypercyclicity, we will need to introduce another family of operators of C*+*-type in order to construct examples of -frequent hypercyclic operators on which are not frequently hypercyclic. But before moving over to the presentation of this new class of operators, we give in the next subsection a corollary of Theorem 6.22 concerning infinite direct sums of frequently hypercyclic operators.
6.6. Infinite direct sums of frequently hypercyclic operators
A well-known open question, dating back to [5], asks whether the direct sum of a frequently hypercyclic operator with itself has to be frequently hypercyclic. (This is the analogue of Herreros “ problem” for frequent hypercyclicity.) By [32], is hypercyclic as soon as is -frequently hypercyclic. This result of [32] also implies that the direct sum of two (and hence, of infinitely many) -frequently hypercyclic operators is hypercyclic. Indeed, let and be two bounded operators acting respectively on the Banach spaces and , and let , and , be non-empty open subsets of and respectively. By [32], the set has bounded gaps. Also, since is topologically weakly mixing, the set contains arbitrarily long intervals. It follows that is hypercyclic.
Apart from this result of [32], nothing seems to be known concerning this question of [5]. More generally, it seems to be unknown whether the direct sum of two frequently hypercyclic operators is necessarily frequently hypercyclic, or even -frequently hypercyclic. To the best of our knowledge, this question is open even for infinite direct sums of frequently hypercyclic operators. Our aim in this subsection is to use operators of -type on to prove the following result.
Theorem 6.24**.**
Let . There exists a sequence of frequently hypercyclic operators on such that the -sum operator acting on is not -frequently hypercyclic.
The proof of Theorem 6.24 relies on Theorem 6.22, combined with an elementary lemma providing an upper bound for the parameter of a direct sum operator:
Lemma 6.25**.**
Let and be two bounded operators acting respectively on the Banach spaces and . Fix , and let be the -sum operator of and , acting on . Moreover, assume that is hypercyclic. Then .
Proof of Lemma 6.25.
Let be such that for a comeager set of vectors of . For any such vector , with and , there exists a subset of with such that \bigl{(}\|T^{n}_{1}x_{1}\|^{p}+\|T_{2}^{n}x_{2}\|^{p}\bigr{)}^{1/p}<1 for every . Thus for a comeager subset of vectors , so that ; and likewise, . Hence for any as above, which proves the lemma. ∎
As an easy consequence of Lemma 6.25, we obtain
Lemma 6.26**.**
For each , let be a bounded operator on a Banach space . Fix , and let be the -sum of the operators , , acting on the space . Moreover, assume that is hypercyclic. Then
[TABLE]
Proof of Lemma 6.26.
It follows directly from Lemma 6.25 that
[TABLE]
so we only have to prove that
[TABLE]
If the infimum is equal to [math], there is nothing to prove. Suppose that and consider a number such that . Let be a non-empty open subset of . There exist an integer and, for every , a non-empty open subset of , such that contains the set . Since , there exist a vector and a subset of with such that \bigl{(}\sum_{i=1}^{n}\|T_{i}^{k}x_{i}\|^{p}\bigr{)}^{1/p}<1 for every . Setting , we deduce that is a vector of which satisfies . The set of vectors such that is thus dense in , and the usual argument shows that this set is in fact comeager in . It follows that , from which we deduce that . This concludes the proof of Lemma 6.25. ∎
Proof of Theorem 6.24.
Let be a sequence of operators satisfying the assumptions of Theorem 6.22. Denoting for each by , , and the sequences of parameters associated to the operator , we suppose that
[TABLE]
for every . By Theorem 6.22, all the operators are frequently hypercyclic. On the other hand, it follows from (33) that
[TABLE]
so that tends to [math] as tends to infinity. By Lemma 6.26, the operator (which is hypercyclic) satisfies . Hence is not -frequently hypercyclic. ∎
6.7. Operators of C*+,2*-type: UFHC does not imply FHC
In this subsection, we impose the following restrictions on the parameters , , and of a C*+*-type operator : for every ,
[TABLE]
where and are two strictly increasing sequences of integers, satisfying for every . We call operators satisfying these conditions operators of C+,2-type. Observe that we still have with this definition
[TABLE]
This choice of weights is motivated by the differences between the assumptions of Theorem 5.14 and those of Theorem 5.31. If we assume for simplicity that for every and if we consider the vector , we observe that is close to and that the orbit of follows the orbit of during an interval of time proportional to . In view of Theorem 5.14, it thus seems likely that will be -frequently hypercyclic. However, if , it also seems plausible that will not satisfy the assumptions of Theorem 5.31, and hence will possibly not be frequently hypercyclic.
6.7.1. How to be FHC or UFHC
In this subsection, we present some sufficient conditions for an operator of C*+,2*-type to be -frequently hypercyclic.
Theorem 6.27**.**
Let be an operator of C+,2-type on .
- (1)
If \displaystyle\limsup_{k\to\infty}\ \bigl{(}\delta^{(k)}-\tau^{(k)}\bigr{)}=\infty, then is chaotic. 2. (2)
If , then is chaotic and -frequently hypercyclic. 3. (3)
If , then is chaotic and frequently hypercyclic.
Démonstration.
The proofs of assertions (1) and (3) are completely similar to the ones given in the proof of Theorem 6.17. As for (2), it is a consequence of Theorem 6.7. Let us fix
[TABLE]
Fix also , and an integer . There exists an integer such that
[TABLE]
We then have
[TABLE]
and, for every :
[TABLE]
Applying Theorem 6.7 with , it follows that is -frequently hypercyclic. ∎
We see from this proof that the new structure of the weights , , compared with the case of operators of C*+,1*-type, allows us to provide different conditions for -frequent hypercyclicity and frequent hypercyclicity of operators of C*+,2*-type. Our next result gives, under some additional assumptions, a necessary and sufficient condition for frequent hypercyclicity of operators of C*+,2*-type which, combined with (2) of Theorem 6.27, will ultimately allow us to construct -frequently hypercyclic operators of C*+,2*-type which are not frequently hypercyclic.
Theorem 6.28**.**
Let be an operator of C+,2-type on , and for every , set \gamma_{k}:=2^{\,\delta^{(k-1)}-\tau^{(k)}}\bigl{(}\Delta^{(k)}\bigr{)}^{1-\frac{1}{p}}. Suppose that the sequence is non-increasing and that it satisfies the following three conditions:
[TABLE]
Then is frequently hypercyclic if and only if
Démonstration.
Since , the condition is equivalent to
[TABLE]
and by assertion (3) of Theorem 6.27, this condition implies that is frequently hypercyclic.
The proof of the converse assertion proceeds exactly as in the proof of the implication in Theorem 6.18. Suppose that , and let be a hypercyclic vector for . Our aim is to prove that cannot be a frequently hypercyclic vector for , using Theorem 6.10. We set
[TABLE]
This sequence is non-increasing and satisfies . Setting
[TABLE]
we have .
Also, by Fact 6.12, we have for every , every , and every , where , that
[TABLE]
and
[TABLE]
At this point, we diverge from the proof of Theorem 6.18, and set for every and every . We then have for every and every . In order to check that assumption (C’) of Theorem 6.10 holds true, we use again Fact 6.13. We have for every and
[TABLE]
It follows from Fact 6.13 that
[TABLE]
for every , every , and every . Now, we have for every , so that .
For every , we have
[TABLE]
Hence
[TABLE]
since and . Theorem 6.10 thus implies that is not frequently hypercyclic ∎
6.7.2. UFHC but not FHC
As a direct consequence of Theorems 6.27 and 6.28, we now obtain
Theorem 6.29**.**
Let be an operator of C+,2-type on . For every , set \gamma_{k}:=2^{\,\delta^{(k-1)}-\tau^{(k)}}\bigl{(}\Delta^{(k)}\bigr{)}^{1-\frac{1}{p}}. Suppose that the sequence is non-increasing, and that the following conditions are satisfied:
[TABLE]
Then is -frequently hypercyclic but not frequently hypercyclic.
Here is a concrete example, the proof of which is left to the reader.
Example 6.30**.**
If we consider the operator of C*+,2*-type associated to the parameters
[TABLE]
where is a sufficiently large integer, then is -frequently hypercyclic but not frequently hypercyclic.
6.8. Operators of C2-type: chaos plus mixing do not imply UFHC
In this subsection, we introduce yet another class of C-type operators on .
We consider increasing sequences , , , and of integers such that for every , and we denote by the partition of into consecutive finite intervals defined as follows: and for every .
We then require that:
for every ; 2. -
the blocks , , all have the same size ; 3. -
the sequence is given by
[TABLE] 4. -
If belongs to the interval with and , then the weight is given by
[TABLE]
If these conditions are met, we call an operator of C2-type.
Note that the family of C2-type operators contain that of C*+,2*-type operators. Indeed, if we consider the sequences and given by and for every , we get back the definition of C*+,2*-type operators. The specificity of these new operators lies in the fact that for every and every , the set contains integers for which , and, for each of these integers, the central block of weights is translated in a similar way.
6.8.1. How to be topologically mixing
By using Proposition 6.14, we can show that under suitable assumptions on the sequences and , an operator of C2-type can be topologically mixing.
Proposition 6.31**.**
Let be an operator of C2-type on . Suppose that the following two conditions hold true: , and the set
[TABLE]
is cofinite. Then is topologically mixing.
Démonstration.
With the notations of Proposition 6.14, we have to show that the set is cofinite. First of all, observe that the set contains, for every such that , the set . Since tends to infinity as tends to infinity, there exists an integer such that contains the set
[TABLE]
Recalling that
[TABLE]
we deduce that contains the set
[TABLE]
which in its turn contains the set
[TABLE]
(recall that for every integer belonging to ). Fix an integer , and let be an integer of the form , where . Then belongs to , and . It follows that contains every integer of the form , where , i.e. that
[TABLE]
Since the set on the left hand side is cofinite by assumption, the desired result follows from Proposition 6.14. ∎
6.8.2. How not to be UFHC
We now give some conditions ensuring that a C2-type operator fails to be -frequently hypercyclic. This will rely on the following more general version of Fact 6.13.
Fact 6.32**.**
Let be an operator of C-type on , and let . Fix , and define
[TABLE]
Suppose that there exist three integers such that
[TABLE]
Then we have for every
[TABLE]
Démonstration.
The proof is similar to the proof of Fact 6.13 except that the sets , , have to be defined as follows:
[TABLE]
We can then deduce that there exists an integer such that for every :
[TABLE]
Now, we remark that if we set for every , we have
[TABLE]
This particular structure of the sets implies that
[TABLE]
and this yields the desired result. ∎
Proposition 6.33**.**
Let be an operator of C2-type on , and define as usual a sequence by setting \gamma_{k}=2^{\,\delta^{(k-1)}-\tau^{(k)}}\bigl{(}\Delta^{(k)}\bigr{)}^{1-\frac{1}{p}} for every . Suppose that the sequence is non-increasing, and that the following two conditions are satisfied:
[TABLE]
Then is not -frequently hypercyclic.
Démonstration.
Let be a hypercyclic vector for . Our aim is to prove that cannot be a -frequently hypercyclic vector for , using Theorem 6.10. We set
[TABLE]
This sequence is non-increasing and satisfies . Setting
[TABLE]
we have and, by Fact 6.12,
[TABLE]
for every , and
[TABLE]
for every and with belonging to . Setting for every and every , we deduce that
[TABLE]
for every and every . In order to check that assumption (C) of Theorem 6.10 holds true, we use Fact 6.32. Let belong to , and define . We have for every and
[TABLE]
It follows from Fact 6.32 that
[TABLE]
for every , every , and every . For every , we thus have
[TABLE]
Since , it follows that
[TABLE]
By Theorem 6.10, is not -frequently hypercyclic. ∎
6.8.3. Chaotic and mixing operators which are not UFHC
From Proposition 6.31 and 6.33, we immediately deduce
Theorem 6.34**.**
Let be an operator of C2-type on , and for every , set \gamma_{k}=2^{\,\delta^{(k-1)}-\tau^{(k)}}\bigl{(}\Delta^{(k)}\bigr{)}^{1-\frac{1}{p}}. Suppose that the sequence is non-increasing and that the following three conditions are met:
[TABLE]
If the set
[TABLE]
is cofinite, then is chaotic and topologically mixing but not -frequently hypercyclic.
Here is a concrete example of such a chaotic and topologically mixing operator which is not -frequently hypercyclic.
Example 6.35**.**
Let be a positive integer. Consider the operator of C2-type on associated to the parameters
[TABLE]
If is sufficiently large, then is chaotic and topologically mixing but not -frequently hypercyclic.
Démonstration.
The operator is well-defined since
[TABLE]
for every . Moreover, the set
[TABLE]
is cofinite since if is large enough. Indeed,
[TABLE]
and the quantity on the right hand side tends to infinity as tends to infinity. Finally, since and , we have for every . It follows that the remaining assumptions of Theorem 6.34 are also satisfied if is sufficiently large. ∎
Remark 6.36**.**
The operators constructed in [39] (which are chaotic and not -frequently hypercyclic) are never topologically mixing. Indeed, the parameters in this construction satisfy the following three conditions:
the quantity tends to [math] as tends to infinity;
- -
for every , is a multiple of ;
- -
for every , .
Therefore, for any such operator , there exists an integer such that
[TABLE]
It follows that for every integer such that does not belong to the set , does not belong to . Indeed, we have for any
if ;
- -
if belongs to since does not belong to ;
- -
if , by Fact 6.12.
Hence, if is any vector of the unit ball , the vector satisfies
[TABLE]
Thus does not belong to the ball . Since there are infinitely many integers such that does not contain , this shows that is indeed not topologically mixing.
6.9. C-type operators with few eigenvalues
In this subsection, we exhibit a class of C-type operators having only countably many unimodular eigenvalues. This provides further examples of hypercyclic operators on with only countably many unimodular eigenvalues, and such that the associated unimodular eigenvectors span the space (as mentioned previously, the question of the existence of such operators was raised by Flytzanis in [22]).
The general idea of the forthcoming construction is the following: if is an operator of C-type on and if the sequence decreases extremely fast, then the unimodular eigenvalues of must be roots of unity. This is not such a surprising statement if one considers what happens in the “degenerate” case where the sequence is identically equal to [math]: indeed, in this case the operator has the form where the operators , are finite dimensional cyclic operators satisfying . Thus any eigenvalue of must satisfy for some integer .
Before starting our construction, we determine the spectrum of the operators of C-type on which we consider here:
Fact 6.37**.**
Let be a hypercyclic C-type operator on such that
[TABLE]
Then the spectrum of is the closed disk , where
[TABLE]
If is either a -type or a -type operator on , the spectrum of is thus the closed disk .
Démonstration.
Using the notation employed in the proof of Fact 6.2, we observe that is a compact perturbation of the direct sum operator on . Also, the assumption of Fact 6.37 implies that is itself a compact perturbation of the forward weighted shift on defined by
[TABLE]
So is a compact perturbation of , and it then follows from the Fredholm alternative that if is any element of , then is an eigenvalue of . But as is hypercyclic, its adjoint has no eigenvalue, and it follows that is contained in . Hence, is contained in . Conversely, the same argument shows that any is an eigenvalue of . But the only eigenvalue of is [math], so that is contained in . Now, it is well-known that the is the closed disk (see for instance [46]), so that . Since is closed, it follows that , and this concludes the proof of Fact 6.37. ∎
We now come back to our construction of C-type operators with few unimodular eigenvalues. In order to simplify the expressions involved in the results which we are about to state, we adopt the following notation: if is an operator of C-type, we set
[TABLE]
Also, we will say that an increasing sequence of positive integers is a -sequence if
[TABLE]
Theorem 6.38**.**
Let be an operator of C-type on . Assume that for every -sequence , we have
[TABLE]
Then each unimodular eigenvalue of is a root of unity. More precisely, any such eigenvalue must satisfy for some integer .
The condition appearing in Theorem 6.38 may look a bit strange since it involves all -sequences (but it will show up naturally in the proof of Theorem 6.38). However, its general meaning is clear: should go to [math] quite fast as goes to infinity. Here is a consequence of Theorem 6.38 that makes it rather transparent. Recall that if is a C*+,1*-type or a C*+,2*-type operator on , then for every and every .
Corollary 6.39**.**
Let be a C+,1-type or a C+,2-type operator on . Assume that tends to infinity and k\log\Delta^{(k)}=O\bigl{(}\Delta^{(k-1)}\bigr{)} as tends to infinity. If
[TABLE]
then all the unimodular eigenvalues of must be roots of unity.
Proof of Corollary 6.39.
Let be a -sequence, and for each , let us denote by the quantity appearing in (35), namely
[TABLE]
For any , let us denote by the unique positive integer such that belongs to the interval . Then because is a -sequence.
Observe first that the partial products
[TABLE]
remain bounded (in fact, they tend quickly to [math]). Also, since the sequence is bounded, there exists a constant such that
[TABLE]
Since tends to infinity as tends to infinity, it follows that there exists a positive constant such that
[TABLE]
Moreover, there also exists a positive constant such that for every (again because tends to infinity). Lastly, we have
[TABLE]
and since k\log\Delta^{(k)}=O\bigl{(}\Delta^{(k-1)}\bigr{)} the quantity \bigl{(}\Delta^{(k_{m})}\bigr{)}^{k_{m}} is dominated by for some positive constant . Putting things together, and remembering that for every , we obtain that there exists a positive constant such that
[TABLE]
By (36), this concludes the proof of Corollary 6.39. ∎
Remark 6.40**.**
Condition (36) is compatible with those appearing for example in Theorems 6.18, 6.22 or 6.29.
Proof of Theorem 6.38.
We start the proof with the following fact.
Fact 6.41**.**
Fix , and let be a sequence of positive integers tending to infinity such that is a multiple of for every . Assume that for every . Then there exist infinitely many integers such that
[TABLE]
Proof of Fact 6.41.
Writing as with , we have for every
[TABLE]
Since is a multiple of for every and every , it follows that
[TABLE]
So it suffices to show that there exist infinitely many integers such that
[TABLE]
Towards a contradiction assume that there exists an integer and, for every , an integer such that
[TABLE]
We then have
[TABLE]
and since is a multiple of , it follows that
[TABLE]
Since tends to infinity as tends to infinity, we conclude that , so that , which stands in contradiction with our initial assumption. ∎
Let us now show that every unimodular eigenvalue of satisfies for some integer . Let be a unimodular eigenvalue of , and towards a contradiction, assume that for every . Note that since is an even multiple of , we also have for every .
Let be an eigenvector of associated to the eigenvalue . The equation yields that for every and every , we have
[TABLE]
and
[TABLE]
Since , we deduce from the last identity that for every , there exists with such that
[TABLE]
Thus, if we set
[TABLE]
the following property holds true: for every , there exists with such that
[TABLE]
Let us choose an integer such that (since is non-zero, such an integer does exist by (37)). The argument given just above allows us to construct a -sequence such that
[TABLE]
We then have
[TABLE]
and hence (by (37) again)
[TABLE]
If we now set
[TABLE]
it suffices to prove that does not tend to infinity as tends to infinity in order to obtain a contradiction, since this would imply that does not belong to . In view of our assumption (35), this will be verified if we are able to show that
[TABLE]
At this point, we apply Fact 6.41, considering the sequence defined by setting for every . The conclusion is that there exist infinitely many integers such that
[TABLE]
which gives (39). Theorem 6.38 is proved. ∎
Remark 6.42**.**
Assume that . If is a -sequence such that , the proof of Theorem 6.38 shows that if and if we set
[TABLE]
then is an eigenvector of associated to the eigenvalue , provided that the series defining is convergent. This has the following two interesting consequences.
- (1)
An operator of C-type may quite well have a unimodular eigenvalue such that the associated eigenspace is not one-dimensional. This happens in particular if for some integer and if there exists such that . 2. (2)
If we assume that v_{n(l)}=\Big{(}\prod_{\nu=b_{n(l-1)}+1}^{b_{n(l-1)+1}-1}w_{\nu}\Big{)}^{-1}, we obtain the following expression for :
[TABLE]
This vector is well-defined for some unimodular numbers which are not roots of unity, provided that the sequence is suitably chosen. Therefore, operators of C-type can have unimodular eigenvalues which are not roots of unity.
6.10. C*+,1*-type operators can be ergodic
To conclude this section, we show that C*+,1*-type operators can also have lots of eigenvalues, to the point of being even mixing in the Gaussian sense.
Theorem 6.43**.**
Let be a C+,1-type operator on . Suppose that , and that the series
[TABLE]
is convergent. Then is mixing in the Gaussian sense.
Démonstration.
Since is a C*+*-type operator, we know that the map is given by if , . From this it follows that a sequence with is a -sequence if and only if there exists an increasing sequence of positive integers such that
[TABLE]
More precisely, is for each the unique positive integer such that . In this case, we say that is the -sequence associated to the sequence .
With these notations, the formula of Remark 6.42 defining for a -sequence and an element of can be rewritten as follows:
[TABLE]
We will say that a -sequence with is a good -sequence if is well-defined for every and if the map is continuous from into .
Fact 6.44**.**
For any integer , there exists a good -sequence passing through , i.e. such that for some integer .
Proof of Fact 6.44.
Write as with , and set for every . Then the -sequence associated to passes through . Moreover, since for every , it follows from the assumption that the series (40) is convergent that the series
[TABLE]
is convergent as well. Since \Bigl{|}\prod_{l=1}^{m}(1+\lambda^{\Delta^{(k_{l})}})\Bigr{|}\leq 2^{m} for every and every , this implies that is a good -sequence. ∎
Fact 6.45**.**
Let be a good -sequence. If is a vector of which is orthogonal to for every , then
[TABLE]
Proof of Fact 6.45.
Let be the increasing sequence of integers associated to , and set . By assumption and in view of the definition of , we have
[TABLE]
for every , where the are non-zero scalars which do not depend on .
Equality (41) applied to yields that . Indeed, since , we have , and the factor appears in each of the terms \prod_{l=1}^{m}\bigl{(}1+\lambda^{\Delta^{(k_{l-1})}}\bigr{)}. Now, take such that . Then for every . Moreover, is non-zero because is an even multiple of . So we get
[TABLE]
for every such that . Denoting by the set of all such elements , and recalling that , we can rewrite (42) as
[TABLE]
where for every . Since has cardinality , it follows that for every , and hence that for every .
Continuing in this way, we obtain that for every , for every belonging to the interval , which proves our claim. ∎
It is now easy to conclude the proof of Theorem 6.43. Let us denote by the set of all good -sequences, and set for every and every . Then the maps , , are continuous eigenvector fields for defined on . Facts 6.44 and 6.45 then imply that \overline{\rm span}\,\bigl{\{}E_{\mathbf{n}}(\lambda);\;\mathbf{n}\in\mathbf{G}\,,\,\lambda\in\mathbb{T}\bigr{\}}=\ell_{2}(\mathbb{N}). By [8], it follows that is strongly mixing in the Gaussian sense. ∎
Example 6.46**.**
If we consider the C*+,1*-type operator on associated to the data , and , , then is mixing in the Gaussian sense.
Remark 6.47**.**
Condition (40) is incompatible with the “additional assumptions” of Theorem 6.18. Indeed, the latter imply that tends to zero as tends to infinity, and hence that tends to infinity.
7. A few questions
We conclude the paper with a short list of questions. Most of them have already been stated in the paper, at least implicitly. Some of them will perhaps be found interesting and not hopelessly intractable.
Question 7.1**.**
For which Banach spaces with separable dual do the “typicality” results proved here for Hilbert spaces hold true?
Question 7.2**.**
Do there exist ergodic operators on without eigenvalues?
Question 7.3**.**
Do there exist at least Hilbert space operators admitting non-trivial invariant measures but no eigenvalues?
Question 7.4**.**
Let . What is the descriptive complexity of and , with respect to SOT and/or ? In particular, are these sets Borel in ?
Question 7.5**.**
Let . Is a true set in ? Is a true difference of sets?
Question 7.6**.**
Let . Is Borel in ?
Question 7.7**.**
Let . Assume that is hypercyclic, and that there exists a sequence of unimodular eigenvectors with rationally independent eigenvalues spanning a dense linear subspace of . Is ergodic, or at least frequently hypercyclic? -frequently hypercyclic?
Question 7.8**.**
Let be a sequence of distinct unimodular complex numbers tending to , and let be a bounded sequence of positive numbers. If the operator acting on is hypercyclic, is it necessarily ergodic, or at least frequently hypercyclic? -frequently hypercyclic?
Question 7.9**.**
When exactly is an operator of the form hypercyclic?
Question 7.10**.**
Let . Is the class of chaotic operators belonging to comeager in ?
Question 7.11**.**
Let be a Banach space operator. Assume that has a dense set of uniformly recurrent points. Does it follow that has a non-zero periodic point?
Question 7.12**.**
Let be a hypercyclic operator. Assume that and that admits an invariant measure with full support. Does it follow that is -frequently hypercyclic?
Question 7.13**.**
Let be a Banach space, and let have the OSP. Are the ergodic measures for dense in the space of all -invariant measures?
Question 7.14**.**
On which Banach spaces is it possible to construct frequently hypercyclic operators which are not ergodic and/or -frequently hypercyclic operators which are not frequently hypercyclic?
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The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 2[2] S. Bartoll, F. Martínez-Giménez, A. Peris, Operators with the specification property . J. Math. Anal. Appl. 436 (2016), 478–488.
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