Separation conditions on controlled Moran constructions
Antti K\"aenm\"aki, Markku Vilppolainen
TL;DR
This paper extends the equivalence between the open set condition and positive Hausdorff measure from self-similar sets to a broader class of Moran constructions, analyzing various types of these constructions.
Contribution
It establishes a new equivalence for Moran constructions, generalizing known results from self-similar sets and exploring different Moran structures.
Findings
Proves the equivalence for a class of Moran constructions.
Analyzes various types of Moran constructions.
Extends known results from self-similar sets.
Abstract
It is well known that the open set condition and the positivity of the -dimensional Hausdorff measure are equivalent on self-similar sets, where is the zero of the topological pressure. We prove an analogous result for a class of Moran constructions and we study different kinds of Moran constructions with this respect.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Separation conditions on controlled Moran constructions
Antti Käenmäki and Markku Vilppolainen
Department of Mathematics and Statistics
P.O. Box 35 (MaD)
FI-40014 University of Jyväskylä
Finland
Abstract.
It is well known that the open set condition and the positivity of the -dimensional Hausdorff measure are equivalent on self-similar sets, where is the zero of the topological pressure. We prove an analogous result for a class of Moran constructions and we study different kinds of Moran constructions with this respect.
Key words and phrases:
Moran construction, iterated function system, limit set, invariant set, separation condition, open set condition, Hausdorff measure, Hausdorff dimension.
2000 Mathematics Subject Classification:
Primary 28A80; Secondary 37C45.
AK acknowledges the support of the Academy of Finland (project #114821).
1. Introduction
The widely studied class of self-similar sets was introduced by Hutchinson [15]. A mapping is called a similitude mapping if there is such that whenever . If the similitude mappings are contractive, that is, all the Lipschitz constants are strictly less than one, then a nonempty compact set is called self-similar provided that it satisfies
[TABLE]
From this, one can easily see that the set consists of smaller and smaller pieces which are geometrically similar to . However, the self-similar structure is hard to recognize if these pieces overlap too much. Hutchinson [15] used a separation condition which guarantees that we can distinguish the pieces. The idea goes back to Moran [31] who studied similar constructions but without mappings. In the open set condition, it is required that there exists an open set such that all the images are pairwise disjoint and contained in . Lalley [21] used a stronger version of the open set condition. In the strong open set condition, it is required that the open set above can be chosen such that .
Assuming the open set condition, Hutchinson [15, §5.3] proved that the -dimensional Hausdorff measure of is positive, where is the zero of the so-called topological pressure. See also Moran [31, Theorem III] for the corresponding theorem for the Moran constructions. Schief [34, Theorem 2.1] showed, extending the ideas of Bandt and Graf [2], that the open set condition is not only sufficient but also a necessary condition for the positivity of the Hausdorff measure. In fact, he proved that implies the strong open set condition. Later, Peres, Rams, Simon, and Solomyak [33, Theorem 1.1] showed that this equivalence also holds for self-conformal sets. See also Fan and Lau [11], Lau, Rao and Ye [22], and Ye [40]. Observe that in these results, it is essential to have finite number of mappings. Szarek and Wȩdrychowicz [35] have shown that in the infinite case the open set condition does not necessarily imply the strong open set condition.
The main theme in this article is to study the relationship between separation conditions and the Hausdorff measure on limit sets of Moran constructions. More precisely, we study what can be said about Schief’s result in this setting. Since the open set condition requires the use of mappings, we introduce a representative form for it to be used on Moran constructions. We also study invariant sets of certain iterated function systems without assuming conformality. We generalize many classical results into these settings.
The article is organized as follows. In Section 2, we introduce the concept of semiconformal measure on the symbol space and prove the existence of such measures. The projection of a semiconformal measure onto the limit set of a Moran construction gives us valuable information about the limit set provided that the construction is properly controlled and the pieces used in the construction are appropriately separated. We introduce the definitions of the controlled Moran construction (CMC) and suitable separation conditions in Section 3. We also specify a class of CMC’s, the so called tractable CMC’s, for which a natural separation condition is equivalent to the positivity of , where is the limit set and the zero of the topological pressure. In Section 4, we consider a subclass of tractable CMC’s, which we call semiconformal CMC’s. We show that this class has properties that allow us to consider it as a natural replacement of the class of conformal iterated function systems into the setting of Moran constructions. Without the assumption of conformality, we study separation conditions on iterated function systems in Section 5. The last section is devoted to examples.
2. Semiconformal measure
In this section, we work only in the symbol space. We begin by fixing some notation to be used throughout this article. As usual, let be a finite set with at least two elements. Put and . Now for each , there is such that . We call this the length of and we denote . The length of elements in is infinity. Moreover, if and , then with the notation , we mean the element obtained by juxtaposing the terms of and . For and , we define and we call the set a cylinder set of level . If and , we define to be the unique element for which . We also denote . With the notation , we mean that the elements are incomparable, that is, . We call a set incomparable if all of its elements are mutually incomparable. Finally, with the notation , we mean the common beginning of and , that is, , where .
Defining
[TABLE]
for each , the couple is a compact metric space. We call a symbol space and an element a symbol. If there is no danger of misunderstanding, let us call also an element a symbol. Define the left shift by setting
[TABLE]
With the notation , we mean the symbol . Observe that to be precise in our definitions, we need to work with “empty symbols”, that is, symbols with zero length. However, this is left to the reader.
We now present sufficient conditions for the existence of the so-called semiconformal measure. Our presentation here has common points with [5] and [4, §2.1.2]. Suppose the collection satisfies the following two assumptions:
- (S1)
There exists a constant such that
[TABLE]
whenever . 2. (S2)
as .
Given , we define the topological pressure to be
[TABLE]
The limit above exists by the standard theory of subadditive sequences since
[TABLE]
using (S1).
As a function, is convex: Let and . Now Hölder’s inequality implies
[TABLE]
from which the claim follows. According to (S2), we may choose so that . Then, using (S1), we have
[TABLE]
Hence as and, noting that , there exists a unique for which .
Lemma 2.1**.**
Suppose . Then
[TABLE]
whenever .
Proof.
Since
[TABLE]
by (S2), we get
[TABLE]
for each . The other inequality follows similarly from (2.1). ∎
Let be the linear space of all bounded sequences on the real line. Recalling [32, Theorem 7.2], we say that the Banach limit is any mapping for which
- (L1)
is linear, 2. (L2)
L\bigl{(}(x_{n})_{n\in\mathbb{N}}\bigr{)}=L\bigl{(}(x_{n+1})_{n\in\mathbb{N}}\bigr{)}, 3. (L3)
\liminf_{n\to\infty}x_{n}\leq L\bigl{(}(x_{n})_{n\in\mathbb{N}}\bigr{)}\leq\limsup_{n\to\infty}x_{n}.
To simplify the notation, we denote \operatorname*{Lim}_{n}x_{n}=L\bigl{(}(x_{n})_{n\in\mathbb{N}}\bigr{)}.
We call a Borel probability measure on -semiconformal if there exists a constant such that
[TABLE]
whenever . A Borel probability measure on is called invariant if \mu([\mathtt{i}])=\mu\bigl{(}\sigma^{-1}([\mathtt{i}])\bigr{)} for each and ergodic if or for every Borel set for which . The use of the Banach limit is a rather standard tool in producing an invariant measure from a given measure, for example, see [38, Corollary 1] and [28, Theorem 3.8]. In the following theorem, we construct a family of semiconformal measures by applying the Banach limit to a suitable collection of bounded set functions.
Theorem 2.2**.**
For each there exists a unique invariant -semiconformal measure. Furthermore, it is ergodic.
Proof.
Define for each and
[TABLE]
Letting and using using (L1) and (L2), we have
[TABLE]
whenever . Since, by Lemma 2.1,
[TABLE]
and similarly the other way around, we have, using (L3),
[TABLE]
Next define for each and
[TABLE]
Letting , we have and, using (2.3),
[TABLE]
whenever . Observe also that
[TABLE]
whenever . Using now (2.4) and Lemma 2.1, we have
[TABLE]
and similarly the other way around. Hence
[TABLE]
Now, identifying with the cylinder , we notice, using (2.5), that is a probability measure on the semi-algebra of all cylinder sets. Hence, using the Carathéodory-Hahn Theorem (see [39, Theorem 11.20]), extends to a Borel probability measure on . Observe that by (2.6) and (2.7), is an invariant -semiconformal measure.
We shall next prove that is ergodic. We have learned the following argument from the proof of [28, Theorem 3.8]. Assume on the contrary that there exists a -measurable set such that and . Fix and take an incomparable set for which and
[TABLE]
Using (2.7), we infer
[TABLE]
Therefore
[TABLE]
for each . Denote \gamma=\bigl{(}1-2^{-1}D^{-16t}\mu(I^{\infty}\setminus A)\bigr{)} and . Take an incomparable set for which and . Since now, using (2.8),
[TABLE]
we have finished the proof of the ergodicity.
To prove the uniqueness, assume that is another invariant -semiconformal measure. Now there exists such that whenever . According to the uniqueness of the Carathéodory-Hahn extension, this inequality implies that also . Using the ergodicity of the measure , it follows that , see [37, Theorem 6.10]. The proof is finished. ∎
Let us next prove two lemmas for future reference. Define for
[TABLE]
and
[TABLE]
Lemma 2.3**.**
Suppose is an invariant ergodic Borel probability measure on . Then and for every provided that .
Proof.
Take such that . Notice that and due to the invariance of , it holds that \mu\bigl{(}\sigma^{-1}(I^{\infty}\setminus\Omega_{\mathtt{i}}^{0})\bigr{)}=\mu(I^{\infty}\setminus\Omega_{\mathtt{i}}^{0}). Since , we have and using the ergodicity of , we have either or . Since
[TABLE]
it follows that . The proof is finished. ∎
Assume that has at least three elements. For a fixed , we denote and define
[TABLE]
Lemma 2.4**.**
Suppose . Then for every .
Proof.
Using Theorem 2.2, we denote with the invariant ergodic Borel probability measure on for which
[TABLE]
for a constant whenever . Assume now on the contrary that there is such that . Using Theorem 2.2, we denote with the unique invariant -semiconformal measure on . Observe that there exists a constant such that
[TABLE]
whenever . Notice also that and by Lemma 2.3. Defining , we have for each
[TABLE]
and similarly the other way around. Hence also is invariant and -semiconformal on . From the uniqueness, we infer , and therefore
[TABLE]
This contradiction finishes the proof. ∎
3. Controlled Moran construction
The collection of compact sets with positive diameter is called a controlled Moran construction (CMC) if
- (M1)
as and , 2. (M2)
there exists a constant such that
[TABLE]
whenever , 3. (M3)
there exists such that
[TABLE]
Lemma 3.1**.**
Given CMC, there are constants and such that for all .
Proof.
Using (M3), we find and such that for every . Fix , take and denote , where is the integer part of , for , and . Since now, by (M2),
[TABLE]
the proof is finished. ∎
Using the assumption (M1) and Lemma 3.1, we define a projection mapping such that
[TABLE]
as . It is clear that is continuous. The compact set is called the limit set (of the CMC). We call a Borel probability measure on -semiconformal if there exists a constant such that
[TABLE]
whenever and
[TABLE]
whenever . Observe that in Section 2 we defined a semiconformal measure on . The overlapping terminology should not be confusing as it is clear from the content which definition we use. Furthermore, for each , we set
[TABLE]
provided that the limit exists. It follows straight from the definition that if there exists a -semiconformal measure on then . Recalling Lemma 2.1, the equation gives a natural upper bound for the Hausdorff dimension of , .
The following proposition provides sufficient conditions for the existence of the -semiconformal measure on . We say that a CMC satisfies a bounded overlapping property if \sup_{x\in E}\sup\bigl{\{}\#R:R\subset\{\mathtt{i}\in I^{*}:x\in X_{\mathtt{i}}\}\text{ is incomparable}\bigr{\}}<\infty. Observe that in the proposition the assumption according to which for each and it holds that whenever is essential, see Example 6.2.
Proposition 3.2**.**
Given a CMC, the limit in (3.1) exists and there is a unique such that . Assuming , there exists an invariant ergodic Borel probability measure on and constants such that
[TABLE]
whenever . Denoting , we have for every -measurable . Furthermore, if in addition the CMC satisfies the bounded overlapping property and for each and it holds that whenever then is a -semiconformal measure on .
Proof.
According to (M2) and Lemma 3.1, the collection satisfies (S1) and (S2). The proof of the first claim is now trivial. Suppose and denote with the -semiconformal measure on associated to this collection, see Theorem 2.2. For fixed and take such that and choose to be the smallest integer for which . Denoting and using (M2), we obtain
[TABLE]
which, according to [10, Proposition 2.2(b)], gives the second claim. Here with the notation , we mean the open ball centered at with radius . Furthermore, if the bounded overlapping property is satisfied then the proof of [17, Theorem 3.7] shows that
[TABLE]
whenever provided that for each it holds \mu\bigl{(}[\mathtt{i};\pi^{-1}(X_{\mathtt{h}}\cap X_{\mathtt{k}})]\bigr{)}\leq m(X_{\mathtt{i}\mathtt{h}}\cap X_{\mathtt{i}\mathtt{k}}). This is guaranteed by our extra assumption. Hence
[TABLE]
which finishes the proof of the last claim. ∎
In the definition that follows, we introduce a natural separation condition to be used on Moran constructions. Given CMC, define for
[TABLE]
and if in addition , we set
[TABLE]
It is often useful to know the cardinality of the set . We say that a CMC satisfies a finite clustering property if . Furthermore, if then the CMC is said to satisfy a uniform finite clustering property.
Definition 3.3**.**
We say that a CMC satisfies a ball condition if there exists a constant such that for each there is such that for every there exists a set such that the collection is disjoint. If above can be chosen to be infinity for every then the CMC is said to satisfy a uniform ball condition. Here with the notation , we mean the convex hull of a given set .
We shall next prove that the (uniform) ball condition and the (uniform) finite clustering property are equivalent.
Lemma 3.4**.**
Given a compact and connected set and , there exists points such that the collection of balls \bigl{\{}B\bigl{(}x_{i},(2k)^{-1}\operatorname{diam}(A)\bigr{)}:i\in\{1,\ldots,k\}\bigr{\}} is disjoint and \#\bigl{\{}i\in\{1,\ldots,k\}:B\bigl{(}x_{i},(2k)^{-1}\operatorname{diam}(A)\bigr{)}\cap B\bigl{(}x,(2k)^{-1}\operatorname{diam}(A)\bigr{)}\neq\emptyset\bigr{\}}\leq 2 for every .
Proof.
Choose such that . Denote the line going through and with and define for each a point y_{i}=\bigl{(}1-\tfrac{i}{k}\bigr{)}y_{1}+\tfrac{i}{k}y_{k}\in L. Using the connectedness of , we find for each a point for which the inner product . The proof is finished. ∎
Theorem 3.5**.**
A CMC satisfies the (uniform) ball condition exactly when it satisfies the (uniform) finite clustering property.
Proof.
We shall prove the non-uniform case. The uniform case follows similarly. Assuming the ball condition, take and . Choose for each a point such that the balls are disjoint as . Now clearly
[TABLE]
for every . Hence
[TABLE]
where denotes the -dimensional Hausdorff measure of the unit ball. This shows that the CMC satisfies the finite clustering property.
Conversely, by the finite clustering property, there exists such that for each there is such that whenever . Choose and for fixed and denote the symbols of with , where . We shall define the points needed in the ball condition inductively. Choose to be any point of and assume the points , where , have already been chosen such that the collection of balls \bigl{\{}B(x_{\mathtt{i}_{i}},\delta r):i\in\{1,\ldots,k\}\bigr{\}} is disjoint. Using Lemma 3.4, we find points such that the collection \bigl{\{}B\bigl{(}y_{j},(4n)^{-1}\operatorname{diam}(X_{\mathtt{i}_{k+1}})\bigr{)}:j\in\{1,\ldots,2n\}\bigr{\}} is disjoint. Since, using (M2),
[TABLE]
for every , Lemma 3.4 says also that the balls , , can intersect at most of balls B\bigl{(}y_{j},(4n)^{-1}\operatorname{diam}(X_{\mathtt{i}_{k+1}})\bigr{)}, . Hence, choosing such that B\bigl{(}x_{\mathtt{i}_{k+1}},(4n)^{-1}\operatorname{diam}(X_{\mathtt{i}_{k+1}})\bigr{)}\cap B(x_{\mathtt{i}_{i}},\delta r)=\emptyset for every , we have finished the proof. ∎
It is evident that the bounded overlapping property does not imply the finite clustering property and in Example 6.1, we show that the converse does not hold either. The natural condition according to which \sup_{x\in E,r>0}\sup\bigl{\{}\#R:R\subset\{\mathtt{i}\in I^{*}:X_{\mathtt{i}}\cap B(x,r)\neq\emptyset\;\mbox{and}\;\operatorname{diam}(X_{\mathtt{i}^{-}})>r\}\;\mbox{is incomparable}\bigr{\}}<\infty clearly implies both the bounded overlapping property and the uniform finite clustering property. See also [28, Lemma 2.7]. However, we do not need this condition as under a minor technical assumption, the finite clustering property implies the bounded overlapping property.
Lemma 3.6**.**
If a CMC satisfies the finite clustering property then it satisfies the bounded overlapping property provided that
[TABLE]
for each .
Proof.
Set . Fix and assume that is a finite and incomparable set such that for each . Choose small enough so that and
[TABLE]
According to the assumption, , and hence, for each there exists at least one such that for some . The incomparability of now implies that for distinct . Consequently, . ∎
Let us examine how the Hausdorff measure is related to the ball condition. Bear in mind that the finite clustering property and the ball condition are equivalent.
Theorem 3.7**.**
If a CMC satisfies the uniform finite clustering property, , and is the measure of Proposition 3.2 then there exist constants and such that
[TABLE]
whenever and . Consequently, .
Proof.
Suppose and is the measure of Proposition 3.2. Seeing that \pi^{-1}\bigl{(}B(x,r)\bigr{)}\subset\bigcup_{\mathtt{i}\in Z(x,r)}[\mathtt{i}], we get for fixed and
[TABLE]
which, together with the uniform finite clustering property and the proof of Proposition 3.2, gives the first claim.
The second claim follows immediately from [27, Theorem 5.7]. ∎
Remark 3.8*.*
We remark that in Theorem 3.7, the measure can be replaced with the Hausdorff measure by recalling [10, Proposition 2.2]. In fact, it is sufficient to assume the finite clustering property instead of the uniform finite clustering property to see that is proportional to . Especially, under this assumption, it holds that .
One could easily prove that if is -semiconformal for some then there exists a set with such that . Since this hardly generalizes to the whole set without any additional assumption, we propose the following definition. We say that a CMC is tractable if there exists a constant such that for each we have
[TABLE]
whenever , , and . See Example 6.2 for an example of a nontractable CMC. Comparing the following theorem to [34, Theorem 2.1] and [33, Theorem 1.1], we see that the uniform ball condition is a proper substitute for the open set condition in the setting of tractable CMC’s. See also Example 6.4.
Theorem 3.9**.**
A tractable CMC satisfies the uniform finite clustering property provided that for the unique satisfying .
Proof.
Assume on the contrary that for each there are and such that . For fixed choose so that for some . We define
[TABLE]
and taking arbitrary and for which , we denote and . Finally, pick such that as . Since now for every , we have, according to the assumption, that . Hence
[TABLE]
for each recalling that . Therefore
[TABLE]
where , and
[TABLE]
where is the measure of Proposition 3.2, , and the constant does not depend on or . Since as , we obtain
[TABLE]
for all , which, according to [10, Proposition 2.2(b)], gives
[TABLE]
Since 1=\mu(\Omega_{\mathtt{i}})\leq m\bigl{(}\pi(\Omega_{\mathtt{i}})\bigr{)}\leq 1 by Lemma 2.3, we have, using (3.3) and Proposition 3.2,
[TABLE]
which leads to a contradiction as . ∎
To summarize the implications of the previous theorem, we finish this section with the following corollary.
Corollary 3.10**.**
For a tractable CMC, the following are equivalent:
- (1)
The ball condition. 2. (2)
The uniform ball condition. 3. (3)
, where . 4. (4)
There exist constants and such that
[TABLE]
whenever , , and .
4. Semiconformal Moran construction
In a tractable CMC, we require that the relative positions of the sets , , follow the rule given in (3.2). The only restriction for the shapes of these sets comes from (M2) and (M3). Assuming more on the shape, we are able to prove that the Hausdorff dimension and the upper Minkowski dimension of the limit set coincide and if the uniform ball condition is satisfied then the dimension of the intersection of incomparable cylinder sets is small. We say that a CMC is semiconformal if there is a constant such that
[TABLE]
whenever . This property implies that the limit set is “approximately self-similar”. Observe that (4.1) is equivalent to the existence of a constant for which
[TABLE]
whenever . We notice immediately that a semiconformal CMC is tractable which indicates, for example, that the finite clustering property and the uniform finite clustering property are equivalent.
Let us first introduce natural mappings for a semiconformal CMC.
Lemma 4.1**.**
If a CMC is semiconformal then for each there exists a mapping such that \varphi_{\mathtt{i}}\bigl{(}\pi(\mathtt{h})\bigr{)}=\pi(\mathtt{i}\mathtt{h}) as and
[TABLE]
whenever .
Proof.
Fix and . Take and using Lemma 3.1, choose such that . Now, using (4.2), we have
[TABLE]
On the other hand, choosing such that , we get similarly
[TABLE]
The claim follows now by letting since according to (4.3), we may define a mapping by setting \varphi_{\mathtt{i}}\bigl{(}\pi(\mathtt{h})\bigr{)}=\pi(\mathtt{i}\mathtt{h}) as . ∎
It follows that the measure of Proposition 3.2 is semiconformal on a semiconformal CMC satisfying the finite clustering property in the following sense.
Lemma 4.2**.**
If a semiconformal CMC satisfies the finite clustering property, , and is the measure of Proposition 3.2 then
[TABLE]
whenever . Here , , are the mappings of Lemma 4.1.
Proof.
Since Lemma 4.1 clearly implies that \operatorname{diam}\bigl{(}\varphi_{\mathtt{i}}(E)\bigr{)} is proportional to , the CMC formed by the sets , , has the same topological pressure as the original CMC. Notice that by the finite clustering property. By the uniqueness of the invariant semiconformal measure on , also the semiconformal measures determined by these CMC’s on are the same. Noting that the finite clustering property remains satisfied in the new setting and trivially for each , Lemma 3.6 implies the bounded overlapping property. By the semiconformality, it is evident that for each and it holds that whenever and hence Proposition 3.2 completes the proof. ∎
Using the mappings of Lemma 4.1, we are able to prove that the Hausdorff dimension and the upper Minkowski dimension of the limit set of a semiconformal CMC coincide even without assuming the ball condition.
Theorem 4.3**.**
If a CMC is semiconformal and then and .
Proof.
We may assume that . Let , , be the mappings of Lemma 4.1. Notice that, using (M2), there exists a constant such that
[TABLE]
whenever and . Take , such that , and . Then choose such that \mathtt{h}|_{n}\in Z\bigl{(}C^{-1}\operatorname{diam}(E)^{-1}r\bigr{)}. Since x_{0}=\varphi_{\mathtt{h}|_{n}}\bigl{(}\pi(\sigma^{n}(\mathtt{h}))\bigr{)}, we have
[TABLE]
for every . Hence
[TABLE]
On the other hand, using (4.4),
[TABLE]
whenever . Therefore for each and there is a mapping and a constant such that
[TABLE]
whenever . The claim follows now from [8, Theorem 4]. ∎
The following simple proposition shows the bi-Lipschitz invariance of a semiconformal CMC. Therefore the collection of all semiconformal CMC’s is sufficiently large. Observe that despite of this property the geometry of the limit set may change a lot under a bi-Lipschitz map, see [26, Lemma 3.2].
Proposition 4.4**.**
If is a semiconformal CMC with as a limit set and is a bi-Lipschitz mapping then is a semiconformal CMC with as a limit set.
Proof.
Fix constants such that
[TABLE]
for every . The condition (M1) is clearly satisfied and since a\operatorname{diam}(X_{\mathtt{i}})\leq\operatorname{diam}\bigl{(}h(X_{\mathtt{i}})\bigr{)}\leq b\operatorname{diam}(X_{\mathtt{i}}) as and a\operatorname{dist}(X_{\mathtt{i}},X_{\mathtt{j}})\leq\operatorname{dist}\bigl{(}h(X_{\mathtt{i}}),h(X_{\mathtt{j}})\bigr{)}\leq b\operatorname{dist}(X_{\mathtt{i}},X_{\mathtt{j}}) as , also the conditions (M2), (M3), and (4.2) are satisfied. The proof is finished. ∎
Examining the method used in [34, Theorem 2.1], one is easily convinced by the usefulness of the set of symbols defined by
[TABLE]
as . See also [22, §2] and [33, §3]. Notice that . The constant in (4.5) is somewhat arbitrary. The reader can easily see that any constant strictly larger than would suffice. Let us next prove two technical lemmas.
Lemma 4.5**.**
Given a CMC, the set is incomparable for every . Furthermore, if then
[TABLE]
Proof.
Fix . Observe that if then clearly . Take . If now , it must be since otherwise , which contradicts with the first observation. If then \sigma^{k}(\mathtt{j}),\sigma^{k}(\mathtt{h})\in Z\bigl{(}\operatorname{diam}(X_{\sigma^{k}(\mathtt{i})})\bigr{)} and hence .
To prove the second claim, fix , take , and denote and . Since \mathtt{j}^{\prime}\in Z\bigl{(}\operatorname{diam}(X_{\mathtt{i}^{\prime}})\bigr{)}, we have, using (M2),
[TABLE]
Therefore, according to (M2),
[TABLE]
and
[TABLE]
The proof is finished. ∎
Lemma 4.6**.**
If a semiconformal CMC satisfies the finite clustering property then
[TABLE]
Proof.
Suppose , , are the mappings of Lemma 4.1, , and is the measure of Proposition 3.2. According to Corollary 3.10 and Theorems 3.5 and 3.7, there exists a constant such that for every and
[TABLE]
Fix , take , and denote and . Since and \mathtt{j}^{\prime}\in Z\bigl{(}\operatorname{diam}(X_{\mathtt{i}^{\prime}})\bigr{)}, we have and
[TABLE]
Using Lemma 4.5, we obtain
[TABLE]
for a point whenever . Hence
[TABLE]
Since, on the other hand, we have a constant such that
[TABLE]
using Lemmas 4.2 and 4.5, we conclude
[TABLE]
whenever . ∎
The following theorem generalizes a crucial point of [34, Theorem 2.1] into the setting of CMC’s. See also [22, Theorem 3.3] and [33, §3].
Theorem 4.7**.**
If a semiconformal CMC satisfies the finite clustering property then there are a constant and a symbol such that
[TABLE]
whenever .
Proof.
Using Lemma 4.6, we choose such that . Therefore clearly
[TABLE]
for every . Since it follows immediately from the definition (4.5) that
[TABLE]
we infer
[TABLE]
whenever .
Take next such that and denote and . Let , where , and choose such that \mathtt{k}|_{k}\in Z\bigl{(}\operatorname{diam}(X_{\mathtt{i}^{\prime}\mathtt{h}})\bigr{)}. Since , we have, using (4.6),
[TABLE]
Hence the definition (4.5) yields . Since , we also have . Similarly, changing the roles of and above, we find such that . This implies that
[TABLE]
Since, on the other hand,
[TABLE]
we infer
[TABLE]
[TABLE]
whenever . Therefore, choosing , we have finished the proof. ∎
As a corollary, we notice that for a semiconformal Moran construction, we may choose the balls in the ball condition to be centered at and placed in such manner that also larger collections (than required in the definition) of them are disjoint.
Corollary 4.8**.**
If a semiconformal CMC satisfies the ball condition then there are a constant and a point such that
[TABLE]
whenever . Here , , are the mappings of Lemma 4.1.
Proof.
Assuming that and are as in Theorem 4.7, the claim follows immediately from Theorems 3.5 and 4.7 by choosing . ∎
Compare the following improvement of Lemma 4.2 to [30, Theorem 3.3] and [23, Theorem 1.6].
Proposition 4.9**.**
If a semiconformal CMC satisfies the ball condition then
[TABLE]
whenever . Here , , are the mappings of Lemma 4.1.
Proof.
Let and be as in Theorem 4.7 and define
[TABLE]
According to Theorem 4.7, we have \varphi_{\mathtt{i}}\bigl{(}\pi([\mathtt{h}])\bigr{)}\cap\varphi_{\mathtt{j}}\bigl{(}\pi([\mathtt{h}])\bigr{)}=\emptyset whenever , and hence also
[TABLE]
as . Thus we get
[TABLE]
whenever from which the Lipschitz continuity implies
[TABLE]
Obviously, is a CMC having as a limit set, whereas is contained in the limit set of the subconstruction . Since it is evident that both of these CMC’s satisfy the uniform finite clustering property, Lemma 2.4 and Theorem 3.7 imply that . Consequently, and the proof is finished. ∎
We shall finish this section with the following observation.
Proposition 4.10**.**
Suppose a collection of compact sets with positive diameter satisfies the following four conditions:
- (C1)
* as and ,* 2. (C2)
there exist such that , 3. (C3)
* for every ,* 4. (C4)
there exists a constant such that
[TABLE]
whenever .
Then the collection is a semiconformal CMC.
Proof.
It suffices to prove (M2) and (M3). To show (M2), observe first that the assumptions (C1) and (C3) guarantee the existence of the limit set and the claim in Lemma 4.1 follows from the assumptions (C1), (C3), and (C4). Notice also that the assumption (C2) implies immediately that . Let , , be the mappings of Lemma 4.1. Then
[TABLE]
whenever and it follows that
[TABLE]
for every . Since
[TABLE]
whenever , we get, by (4.7), that
[TABLE]
whenever . On the other hand,
[TABLE]
implies that
[TABLE]
for every . Since
[TABLE]
whenever and , we get, by (4.8), that
[TABLE]
whenever .
Let us then show (M3). Denote as and choose such that
[TABLE]
for every . By the compactness of , the sequence has a converging subsequence. Let be the limit point of such a subsequence. Now for each there is such that and . Since for all , we have, using (C1) and (C3),
[TABLE]
as . The proof is finished by choosing such that . ∎
5. Semiconformal iterated function system
We assume that for each there is a contractive injection defined on an open subset of and that there also exists a closed and nonempty satisfying
[TABLE]
Here the contractivity of means that there is a constant such that
[TABLE]
whenever . The collection is then called an iterated function system (IFS). As shown in [15, §3], an elegant application of the Banach fixed point theorem implies the existence of a unique compact and nonempty set for which
[TABLE]
Such a set is called an invariant set (for the corresponding IFS). As a side note, it is not necessary to require the mappings to be injective to ensure the existence of the invariant set. However, under this additional assumption, it follows from Brouwer’s domain invariance theorem [6, Theorem IV.7.4] that is open whenever is open.
Observe that the set can be chosen to be a closed neighborhood of the invariant set . Indeed, we fix (if , any positive will do) and take
[TABLE]
The validity of (5.1) is then a consequence of the easily proven fact that
[TABLE]
whenever and .
We say that an IFS is tractable if there exists a compact set and a constant such that for each
[TABLE]
whenever and it defines a CMC in this situation, that is, the collection is a CMC. The limit set of such a CMC is clearly . Here as and .
Lemma 5.1**.**
A tractable IFS defines a tractable CMC.
Proof.
Choose a compact set such that the collection is a CMC. Then
[TABLE]
which implies (3.2) and finishes the proof. ∎
Furthermore, we say that an IFS is semiconformal if the invariant set has positive diameter and there are constants , , and for which as and
[TABLE]
whenever . For an interesting class of quasiregular mappings which admit uniform control of the distortion with respect to iteration, the reader is referred to [25].
The following lemma shows that a semiconformal IFS defines a semiconformal CMC. The natural question whether the converse holds raises from Lemma 4.1. Sufficient geometric conditions on the limit set for the positive answer are provided in [1]. See also [36, Example 6.2].
Lemma 5.2**.**
If is a semiconformal IFS and a compact set with positive diameter satisfies for every then is a semiconformal CMC. In particular, a semiconformal IFS is tractable. Furthermore, the mappings , , are the mappings of Lemma 4.1.
Proof.
To be able to use Proposition 4.10, we have to verify the required assumptions (C1)–(C4). Observe first that (C1) is clearly satisfied and the positivity of implies (C2). Notice also that the sets , , are compact with positive diameter. Since for fixed , we have \underline{s}_{\mathtt{i}}\operatorname{diam}(A)\leq\operatorname{diam}\bigl{(}\varphi_{\mathtt{i}}(A)\bigr{)}\leq\overline{s}_{\mathtt{i}}\operatorname{diam}(A) by (5.4), it follows that
[TABLE]
where and . Hence, applying (5.2) to (5.5) several times, we get \operatorname{diam}\bigl{(}\varphi_{\mathtt{i}}(A)\bigr{)}\leq Cs_{i_{1}}\cdots s_{i_{|\mathtt{i}|}}, which implies the assumption (C3). Since (5.5) gives also the assumption (C4), that is
[TABLE]
as , we have finished the proof of the first claim.
The second claim follows from (5.5) by recalling that the invariant set has positive diameter and it satisfies for every . The third claim follows immediately since the collection is a semiconformal CMC. ∎
We say that an IFS satisfies an open set condition (OSC), if there exists a nonempty open set such that
[TABLE]
whenever . See [31, Theorem III] for the motivation of the definition. Adapting terminology from [3], we call any such nonempty open set a feasible set for the OSC. If there is a feasible set intersecting , we say that a strong OSC is satisfied. As an immediate consequence of the definition, we notice that each nonempty open subset and each image of a feasible set is feasible as well. Thus, using the observation (5.3) repeatedly, we see that the OSC is equivalent to the existence of a feasible set . Recall that is the fixed compact -neighborhood of the invariant set. The next lemma shows that this definition of the OSC is equivalent to the more commonly used one, see [15, §5.2].
Lemma 5.3**.**
An IFS satisfies the OSC exactly when there exists a nonempty open set such that
[TABLE]
as and
[TABLE]
as . Furthermore, there exists a feasible set intersecting if and only if there exists a set as above such that .
Proof.
Defining , where is a feasible set for the OSC, we clearly have as . If , it holds that
[TABLE]
for every and hence
[TABLE]
Noting that the other direction is trivial we have finished the proof. ∎
Given IFS, we say that is forwards invariant if as and backwards invariant if as . For we define
[TABLE]
and for a semiconformal IFS we set
[TABLE]
Here the constant is the same as in the definition of the semiconformal IFS. Observe that is backwards invariant.
Proposition 5.4**.**
Suppose a given IFS is semiconformal. If is a feasible set for the OSC then . Furthermore, if there exists a set such that then is feasible.
Proof.
Let be a nonempty open set for which whenever . It follows that and since is open, we get .
Conversely, it suffices to show that as . Suppose contrarily that there are and such that and . Observe that the inverse mapping has a Lipschitz constant for each . According to Kirszbraun’s theorem [12, §2.10.43], there exists a Lipschitz extension for the mapping having the same Lipschitz constant. Since \overline{\varphi}_{\mathtt{i}}\bigl{(}\varphi_{\mathtt{j}}(A)\bigr{)}\subset F_{A}\cup(\mathbb{R}^{d}\setminus\Omega) and , we have
[TABLE]
using (5.4). Changing the roles of and above, we end up with a contradiction. The proof is finished. ∎
We say that a tractable IFS satisfies the ball condition if the corresponding CMC satisfies the (uniform) ball condition. By Lemma 5.2, this defines the ball condition also for a semiconformal IFS. In this case, we may choose to be the invariant set . Observe that if the IFS is semiconformal and there exists an open set such that for each we have
[TABLE]
for any two distinct then the ball condition is satisfied. See Example 6.5. In particular, the OSC implies the ball condition in the semiconformal case. See also [17, Proposition 3.6]. The following theorem says that, in fact, the ball condition and the strong OSC are equivalent. Example 6.4 shows that this is not true for tractable IFS’s.
Theorem 5.5**.**
A semiconformal IFS satisfies the ball condition exactly when .
Proof.
Let us first prove that the ball condition implies . Recall that is the closed -neighborhood of . We may further assume that
[TABLE]
seeing that implies , which gives . It is now sufficient to find a point with .
According to Theorem 3.5 and Corollary 4.8, there exist a point and a constant such that
[TABLE]
whenever and . It is easy to see that the set is dense in . So, in fact, we have
[TABLE]
for each , which in turn implies that
[TABLE]
for each y\in\varphi_{\mathtt{i}}^{-1}\bigl{(}\varphi_{\mathtt{j}}(E)\bigr{)} when . On the other hand, Lemma 5.2 shows that there is a constant such that
[TABLE]
whenever and . Combining the inequalities above gives
[TABLE]
for each and consequently as desired.
Since the other direction follows immediately from Proposition 5.4, the proof is finished. ∎
The following proposition generalizes [34, Corollary 2.3] and [33, Corollary 1.2] into the setting of semiconformal IFS’s. Although the argument used here is similar to the proof of [33, Corollary 1.2], we give the details for the convenience of the reader.
Proposition 5.6**.**
If a semiconformal IFS satisfies the OSC and then the invariant set is the closure of its interior.
Proof.
As the OSC implies the uniform finite clustering property, we have . Hence there exists a constant such that
[TABLE]
see the defining equation (3.1) and Lemma 2.1. Choose the forwards invariant feasible set as in Lemma 5.3 and consider the set
[TABLE]
The facts that and for every and easily lead to the conclusion that whenever . Furthermore, since for each , we have
[TABLE]
Now (5.6) and (5.7) together imply that . This in turn shows that the set
[TABLE]
is empty, being an open set with zero measure. Here with the notation , we mean the closure of a given set . This means that , giving by the uniqueness of the invariant set. The proof is complete. ∎
A similitude IFS, introduced in [15], is the most obvious example of a semiconformal IFS. Suppose that for each there is a mapping and a constant such that
[TABLE]
whenever . Now for a closed ball centered at the origin, we have whenever provided that the radius of is chosen large enough. The collection is then an IFS and we call it a similitude IFS.
The following proposition is a slightly more general result than [3, Theorem 1].
Proposition 5.7**.**
Given a similitude IFS, the set is forwards invariant and feasible for the OSC provided that and is forwards invariant.
Proof.
According to Proposition 5.4, it suffices to show that as . Assume on the contrary that there exist and such that , that is,
[TABLE]
Notice that here can be chosen to be one. Therefore, since and for every , we obtain
[TABLE]
This contradiction finishes the proof. ∎
The following corollary summarizes the main implications shown for a semiconformal IFS. Notice that the topological pressure here is well defined via Lemma 5.2 as it does not depend on the choice of the corresponding forwards invariant set.
Corollary 5.8**.**
For a semiconformal IFS, the following conditions are equivalent:
- (1)
The ball condition. 2. (2)
The open set condition. 3. (3)
The strong open set condition. 4. (4)
, where is the zero of the topological pressure.
6. Examples
In the last section, we illustrate the preceding theory by providing the reader with several examples. We begin by showing that the uniform finite clustering property does not imply the bounded overlapping property.
Example 6.1*.*
The standard Cantor -set can be defined as the invariant set of the similitude IFS formed by the mappings
[TABLE]
on . It is well known that , where , see [9, Theorem 1.14]. Consider now the CMC , where and . This CMC is tractable by Lemma 5.1. The positivity of implies the uniform finite clustering property by Theorem 3.9. However, using the facts and , we infer by induction that for every , where for each . Since the infinite set is incomparable, we conclude that the bounded overlapping property is not satisfied.
Example 6.2*.*
In this example, we give a CMC which shows that the assumption concerning the relative positions of the sets in the last claim of Proposition 3.2 is indispensable. Besides this, it is also an example of a nontractable CMC. Using the mappings , , from the previous example, set
[TABLE]
and for and
[TABLE]
The CMC determined by these squares obviously has the limit , where is the standard Cantor -set. It is equally obvious that the uniform ball condition is satisfied, which, according to Theorems 3.7 and 3.5 and Remark 3.8, implies that the measure of Proposition 3.2 is proportional to , where as in the previous example. Consequently, whenever is one of the line segments , . Especially,
[TABLE]
for incomparable symbols and satisfying and . We have hereby shown that the measure is not -semiconformal. On the other hand, Lemma 3.6 implies that the bounded overlapping property is satisfied, noting that clearly for each . Therefore, the extra assumption in Proposition 3.2 is really needed.
Furthermore, this CMC is not tractable. This can be deduced from the fact that
[TABLE]
but
[TABLE]
for every .
Example 6.3*.*
In this example, we define a class of non-semiconformal tractable IFS’s. Suppose is a finite set and for each there is a mapping such that
[TABLE]
where and . Defining and for each and , we have
[TABLE]
as . It is clear that as , showing that the IFS is not semiconformal. However, by choosing and , we get for every and
[TABLE]
for each . The collection is thus a CMC and consequently, the IFS is tractable.
According to Corollary 3.10, this CMC satisfies the (uniform) ball condition if and only if , where is the limit set and . For related dimension results, see [29], [14], and [13].
Observe also that choosing, for example, , , , and , it is straightforward to see that the ball condition is automatically satisfied.
Example 6.4*.*
Recall that by Corollary 5.8, a semiconformal IFS satisfies the OSC if and only if it satisfies the ball condition. In this example, we show that for a tractable IFS this equivalence is not necessarily true.
In Example 6.3, let us choose , , , and . It is clear that this tractable IFS satisfies the OSC. It can be seen by a straightforward calculation that the uniform finite clustering property fails, implying that the uniform ball condition does not hold. Alternatively, it follows from the observations done in Example 6.3 that the invariant set has Hausdorff dimension provided that the ball condition is satisfied. However, since is clearly a subset of having Hausdorff dimension , this cannot be the case.
We do not know if there exists a tractable IFS satisfying the ball condition but not the OSC.
Example 6.5*.*
At first glance, it seems that for a semiconformal IFS the OSC (especially via Lemma 5.3) is easier to check than the ball condition. However, there are cases when it is much more convenient to consider the ball condition rather than the OSC. In this example, we consider a familiar self-similar set having this property.
We identify and for notational simplicity and we set . Let and let be the similitudes given by the equations
[TABLE]
where and is the complex conjugate of . Notice that the contraction ratio of both mappings is . The invariant set of the IFS is the well known Lévy’s dragon, see [24]. Since , it follows from [34, Theorem 2.1] that the OSC is satisfied and hence by [34, Corollary 2.3 and its proof] and Lemma 5.3, we conclude that a nonempty open set is feasible only if . Because of the intricate structure of the Lévy’s dragon, such an open set is, a priori, virtually impossible to find. However, it is straightforward to find an open set satisfying
[TABLE]
from which the ball condition follows for any corresponding semiconformal CMC. This can be done by choosing to be the interior of the right-angled triangle and looking at the images as with fixed . The calculations for the disjointness of the images are straightforward since the vertices and of belong to the point grid whereas , and only if and . See [7, p. 222] for an illustration and apply the calculations done in the appendix of [20]. Notice that the set above is not feasible since .
Example 6.6*.*
In this example, we note that any conformal IFS is semiconformal. Suppose is a finite set and for each there is a contractive conformal mapping defined on an open set . Assuming there exists a closed and nonempty satisfying
[TABLE]
the collection is an IFS and we call it a conformal IFS. We deduce from the well known bounded distortion principle that each conformal IFS is semiconformal. See, for example, [28, Remark 2.3]. Observe that the converse does not necessarily hold. For example, the semiconformal IFS constructed in [18, Example 2.1] is not conformal.
Example 6.7*.*
Observe that any IFS conjugated in a bi-Lipschitz way to a conformal IFS is semiconformal. Although the bi-Lipschitz conjugacy preserves positivity and finiteness of the Hausdorff measure, the following example is of special interest as it emphasizes the fact that the use of differentiable mappings is not a necessity in order to prove Corollary 5.8.
We set to be the graph of a nondecreasing continuous function satisfying and . A well known nondifferentiable example of this kind of function is , where is the standard -Cantor set and . In this case, the set is known as Devil’s stairs. We set , , and to be the orthogonal projection onto . Now the mapping is clearly bi-Lipschitz and defining a mapping by setting g(x)=x+f\bigl{(}\operatorname{proj}_{L}(x)\bigr{)}-\operatorname{proj}_{L}(x) for each , the reader can easily see that also is bi-Lipschitz.
Since the line segment is clearly the invariant set of the similitude IFS , where and , the set is the invariant set of a semiconformal IFS . Here is chosen so large that the mappings are contractions.
Devil’s stairs also provides the reader with an example of a semiconformal IFS which is not conformal, see [16, Theorem 2.1].
Example 6.8*.*
Defining for , , and
[TABLE]
we say that a bounded set is uniformly porous if there are and such that for all and . The notion of porosity has arisen from the study of dimensional estimates related to the boundary behavior of various mappings.
Following the proof of [19, Theorem 4.1], we notice that a uniformly porous set is contained in a limit set of a semiconformal CMC satisfying the uniform ball condition such that , see Theorem 3.7.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] P. Alestalo, D. A. Trotsenko, and J. Väisälä. The linear extension property of bi-Lipschitz mappings. Sibirsk. Mat. Zh. , 44(6):1226–1238, 2003.
- 2[2] C. Bandt and S. Graf. Self-similar sets 7. A characterization of self-similar fractals with positive Hausdorff measure. Proc. Amer. Math. Soc. , 114(4):995–1001, 1992.
- 3[3] C. Bandt, N. V. Hung, and H. Rao. On the open set condition for self-similar fractals. Proc. Amer. Math. Soc. , 134(5):1369–1374, 2006.
- 4[4] L. Barreira. A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems. Ergodic Theory Dynam. Systems , 16(5):871–927, 1996.
- 5[5] R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture notes in mathematics , 470, 1975.
- 6[6] A. Dold. Lectures on Algebraic Topology . Springer-Verlag, Berlin, 1972.
- 7[7] G. A. Edgar. Classics on Fractals . Addison-Wesley Publishing Company, New York, 1993.
- 8[8] K. Falconer. Dimensions and measures of quasi self-similar sets. Proc. Amer. Math. Soc. , 106(2):543–554, 1989.