Siegel disks of the tangent family
Weiwei Cui, Hongming Nie

TL;DR
This paper investigates the properties of Siegel disks in tangent family functions, establishing conditions for unboundedness and constructing examples with bounded disks using quasiconformal surgery.
Contribution
It provides a new criterion linking boundary asymptotic values to unbounded Siegel disks and demonstrates construction methods for bounded disks.
Findings
Unbounded Siegel disks contain boundary asymptotic values.
A forward invariant Siegel disk is unbounded iff it contains an asymptotic value.
Constructed examples of functions with bounded Siegel disks.
Abstract
We study Siegel disks in the dynamics of functions from the tangent family. In particular, we prove that a forward invariant Siegel disk is unbounded if and only if it contains at least one asymptotic value on the boundary. Our argument is elementary and function-theoretic. Moreover, by using quasiconformal surgery we also construct functions in the above family with bounded Siegel disks.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Analytic and geometric function theory · Geometric and Algebraic Topology
Siegel disks of the tangent family
Weiwei Cui and Hongming Nie
Abstract
We study Siegel disks in the dynamics of functions from the tangent family. In particular, we prove that a Siegel disk is unbounded if and only if the boundary of its image contains at least one asymptotic value. Moreover, by using quasiconformal surgery we also construct functions in the above family with bounded Siegel disks.
Mathematics Subject Classification: 30D05 (primary), 37F10 (secondary).
Keywords: Meromorphic functions, tangent family, Siegel disks, quasiconformal surgery.
1 Introduction
Let be a transcendental meromorphic function in the plane. The Fatou set of is the set of points whose iterates are defined and form a normal family in the sense of Montel. Its complement on the sphere is the Julia set. The Fatou set is open and each component is called a Fatou component. Periodic Fatou components fall into five possible categories: attracting domains, parabolic domains, Siegel disks, Herman rings and Baker domains. The last possibility never happens for rational functions, but may occur in the transcendental setting. For more details about the Fatou and Julia sets, we refer to [Ber93] and [Mil06].
The dynamical behaviours of a meromorphic function are, in some sense, determined by the iterative properties of its singular values. A singular value of is a point near which at least one branch of the inverse is not well defined. The set of singular values of , denoted by , coincides with the set of critical values and asymptotic values. Recall here that a critical value is the image of a critical point which has vanishing derivative and an asymptotic value is the limit of the image of a curve tending to infinity. Note that we are including here as a possible singular value. For detailed connections between various Fatou components and the singular set, see [Ber93, Ber95, BFJK20] for instance.
On a Siegel disk of period , the iteration is conjugate to an irrational rotation. So Siegel disks contain no critical points. On their boundaries the forward orbits of singular values are dense. However, the interplay between the singular values and the boundaries of Siegel disks is not well understood and it has attracted a lot of interest in recent years (see for example [Her85, Rog98, PZ04, Zha11, CR16, SY16] for rational functions and [Rem04, Rem08, Zha08, BF10, Zak10, BF18, CE18] for transcendental entire functions).
In this short paper, we study Siegel disks in the tangent family
[TABLE]
Each map has exactly two asymptotic values and no critical values. Moreover, the map possesses neither Herman rings, Baker domains nor wandering domains, see [DK89], [Ber93, Corollary 4] and [KK97, Proposition 5.10]. If has a Siegel disk, the symmetry and the orbits of the asymptotic values imply that it has no other types of periodic Fatou components, and hence each of its Fatou components is simply connected. For more dynamics of , we refer to [DK88, DK89, BKL92, KK97]. We mention here that the family is a paradigm of the class in [FK20] consisting of transcendental meromorphic functions with finitely many singular values for which is not an asymptotic value.
Unbounded Siegel disks. An irrational number is of bounded type if the coefficients in its continued fraction expansion are bounded. A result of Graczyk and Światek [GS03] asserts that if a holomorphic function has a Siegel disk which is properly contained in the domain of holomorphy and has bounded type rotation number, then the Siegel disk has a critical point on the boundary. It immediately implies that every Siegel disk of with rotation number of bounded type is unbounded. Indeed, note that is holomorphic away from its poles. Suppose that had a bounded Siegel disk with rotation number of bounded type. Then this Siegel disk would be compactly contained in the complement of the poles and hence its boundary would contain a critical point. This is impossible since has no critical points.
For unbounded Siegel disks of the tangent family, we prove the following.
Theorem 1.1**.**
Suppose that has a Siegel disk . Then is unbounded if and only if .
We remark here that there is no restriction on the period of the Siegel disk in our Theorem 1.1. To see the existence of Siegel disks of higher periods, consider the hyperbolic components (cf. “shell components” in [FK20]) in . Note that for any , there exist hyperbolic components, each element of which has exactly one attracting cycle of period , and hyperbolic components, each element of which has exactly two distinct symmetric attracting cycles of period , see [KK97, Section 8]. Let be such a component. Then is simply connected, see [KK97, Proposition 8.5]. Considering the multiplier map on , as functions in move to , we have that the attracting cycles degenerate to indifferent cycles. Applying [FK20, Theorem 6.13], we in fact obtain functions in possessing the aforementioned cycles whose multipliers are with Brjuno irrational numbers . Moreover, those cycles have the same period as the attracting cycles for functions in ; for, otherwise, they would be parabolic. It follows immediately that such functions are linearizable at those cycles, and hence they have Siegel disks of corresponding periods.
One direction of Theorem 1.1 follows easily from the fact that both asymptotic values are actually omitted: if and is bounded, then there exists a finite point in whose image is an asymptotic value, which is impossible. The reverse implication is similar to Rempe’s result [Rem04] where he proved that an unbounded Siegel disk in the exponential family contains the finite asymptotic value on the boundary of its orbit. To prove the above result, we suppose by contrary that . Then consider two suitable disjoint closed disks around the two asymptotic values. The preimage of the complement of these disks is a horizontal strip containing the real axis. By constructing simple curves connecting the boundaries of the two disks and considering their preimages, we can reach a contradiction due to the unboundedness of .
A transcendental meromorphic function with two singular values is of the form , where is Möbius and is linear (equivalently, one can also say that is of the form ), see Proposition 2.3. Combining our method here with Rempe’s method in [Rem04], one can actually obtain the following theorem. Indeed, if has two singular values, at least one of them is finite. Our method deals with the case that both singular values are finite, while Rempe’s method works in the remaining case.
Theorem 1.2**.**
Let be a transcendental meromorphic function with exactly two singular values. Suppose that has a Siegel disk of period . Then is unbounded if and only if .
Bounded Siegel disks. For the existence of bounded Siegel disks, we construct a map in having a bounded Siegel disk with quasicircle boundary. Using quasiconformal surgery, we show
Theorem 1.3**.**
There exists such that has a bounded Siegel disk around [math] with quasicircle boundary and .
2 Unbounded Siegel disks
In this section, we prove that the boundary of the image of an unbounded Siegel disk for must intersect with the singular set. We will use the notations (resp. ) for the open (resp. closed) euclidean disk of radius around . Let and write for simplicity. For , put and . Moreover, for , denote the horizontal strip of width symmetric with respect to the real axis.
First we prove the following basic result for .
Proposition 2.1**.**
Let be a domain in . If , then is contained in for some .
Proof.
By assumption, for some we can choose two disks and with disjoint closures such that for . It is clear that for each , the domain is an upper or lower half plane. This can be seen by considering , where and . Therefore, there exists (depending only on ) such that . In particular, this implies that . ∎
Now we prove Theorem 1.1.
Proof of Theorem 1.1.
As mentioned in Section 1, it suffices to show that if is unbounded, then . Suppose to the contrary that . Note that . Fix sufficiently small and set
[TABLE]
such that and . Define
[TABLE]
Since , the map is a covering map. From the proof of Proposition 2.1, there exists such that .
Let and be Riemann maps such that and , where is the center of . It follows that is an automorphism of fixing [math]. Hence there exists with such that
[TABLE]
Pick and consider . It follows that and is a Jordan curve. Let and be two points. Then there exists a simple curve connecting and such that and contains exactly two points. Indeed, to obtain such a , we take the image of a diameter line in under , and then continue it in until reaching and . See Figure 1.
Now consider the preimage . There exists a bounded simple curve connecting both boundaries of such that with . It follows that for , and hence is injective on . Set . Then and . Without loss of generality, we can assume . By the choice of , we have . Then for all since is injective on .
Since is unbounded, there exists such that
[TABLE]
where . Without loss of generality, we can assume . It follows that . Set . Note that and is a Jordan curve. We can choose a simple curve connecting and such that contains exactly two points and
[TABLE]
Indeed, to obtain such a , we continue in until reaching points in , and then continue it in until reaching and .
For the preimage , there exists a bounded simple curve connecting both boundaries of such that and with . Then and for . Moreover, is injective on .
We claim that if . It suffices to show the claim holds for since . Suppose that there exists such that . It follows that since . Pick and note that for all ,
[TABLE]
We have that . It is impossible. Thus the claim holds.
To finish the proof, we can derive a contradiction by locating . The above claim implies that is contained in the interior of the quadrilateral bounded by the and the two boundaries of . The choice of and the properties of guarantee that . However, note that by the choice of . Since is injective on , we have . It is a contradiction. ∎
Remark 2.2*.*
In principle, the only requirements of in the proof of Theorem 1.1 are that is simply connected and is univalent in . In particular, if is a simply connected Fatou component of in which is univalent, Theorem 1.1 holds.
Our next result concerns the form of transcendental meromorphic functions with exactly two singular values, which, as mentioned in Section 1, is used to prove Theorem 1.2. It seems to be classical but is not easy to locate a reference. The idea of our proof is suggested by Walter Bergweiler. For a slightly different proof, see [AC20, Theorem 2.1].
Proposition 2.3**.**
Let be transcendental and meromorphic with exactly two singular values. Then is of the form , where is Möbius and is linear.
Proof.
Post-composing a Möbius transformation, we can assume that the singular values of are at [math] and . Then can be continued analytically in the whole plane and thus equals to some entire function . It follows that . Thus, to show the existence of , it suffices to prove that is linear.
Note that , and both [math] and are asymptotic values of . Then has at least two asymptotic values at [math] and . Since has exactly two singular values, it follows that . Note that is non-constant. By the little Picard theorem, the map omits at most one point in . If has no omitted values (i.e., is onto), then will omit [math] and as does. If has exactly one omitted value, say . Then is equal to either [math] or , since otherwise , i.e., , will have a third singular value. By considering if necessary, we may assume that . Then will be omitted by . This is because any pole of will otherwise have a preimage under , which is impossible since omits . It follows that is an entire function. By the Denjoy-Carleman-Ahlfors theorem, the order of is at least . So is a polynomial; for, otherwise, a result of Pólya [Pol26], which states that for two entire functions and , the composition has infinity order unless is of finite order and is a polynomial or has order zero and is of finite order, implies that will have infinity order, while has order . Since is locally univalent in the whole plane, we in fact have that is linear. This completes the proof. ∎
Remark 2.4*.*
Proposition 2.3 implies that there does not exist a meromorphic function with exactly one critical value and one asymptotic value.
3 Bounded Siegel disks
In this section, we prove Theorem 1.3. It is a standard application of quasiconformal surgery. For similar applications, we refer to [Rem03] for the exponential family and [Zha08] for the sine family.
Proof of Theorem 1.3.
Let be an irrational number of bounded type and set . Then has an (unbounded) Siegel disk around [math]. On , the map is conjugate to an irrational rotation by a biholomorphic map . Since , the Siegel disk is symmetric about [math] and hence is odd. Fix some and set . It follows that is symmetric about [math].
Consider the Riemann map fixing . Note that . Then extends to a map on , see [BK87, Theorem A]. Moreover, the map is odd since is symmetric about [math]. Furthermore, the map extends to an odd quasiconformal map on , see [BF14, Proposition 2.30], which we still call the extension . On , define
[TABLE]
It follows that is a circle diffeomorphism. By a result of Herman [Her86], also see [BF14, Theorem 3.21], there exists such that is quasisymmetrically conjugate to an irrational rigid rotation , but not conjugate. Denote this quasisymmetric conjugacy by and post-compose with a rotation such that . Following from the proof of [Zha08, Lemma 2.6], we have that the map is odd. Let be the Douady-Earle extension of , see [DE86]. Then is also odd. Define
[TABLE]
Now we pull back the standard complex structure by and obtain a complex structure in denoted by . Let be the complex structure on defined as follows. For , if the forward -orbit of intersects with , then is the pull back of along the orbit. Otherwise, put . Then is -invariant and . We claim that is bounded. Indeed, note that . Then any point whose -orbit intersects with is either in or in . From the construction and the fact that is holomorphic on , we know that only depends on and hence has a uniform upper bound on . Thus the claim holds.
Let be a quasiconformal homeomorphism of the Riemann sphere which solves the Beltrami equation given by and fixes [math] and , see [Ahl06]. It follows that is odd, see [Zha08, Lemma 2.10]. Now set . Then is a transcendental meromorphic function having a Siegel disk centred at [math] and . We claim that , and hence is a quasicircle and contains no asymptotic values. Suppose . Then the map extends to a conformal map on . Since , it follows that is analytically conjugate to . By the definition of , we know that is analytic on . Hence is analytically conjugate to . This contradicts with the choice of .
Now we claim that there exists such that . Note that from the construction, the map is a meromorphic function with exactly two singular values such that and . Then there exist a Möbius transformation and a linear map such that . Note that for and . We can assume for some . Since , we have is odd. It follows that either or for some . Since , we have in fact . The claim holds.
Since , set and . Then . Note that . The map has a Siegel disk around [math]. Moreover, the multiplier of at the fixed point [math] is . It follows that . Set . Then is a desired map. ∎
Acknowledgement*.*
The authors thank Weixiao Shen, Gaofei Zhang and Jianhua Zheng for useful discussion and helpful comments. The authors also thank the referees for careful reading and helpful suggestions. The first author was partially supported by the China Postdoctoral Science Foundation (No. 2019M651329). The second author was partially supported by ISF Grant 1226/17.
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