Bowen Parameter and Hausdorff Dimension for Expanding Rational Semigroups

Bowen Parameter and Hausdorff Dimension for Expanding Rational Semigroups

Hiroki SUMI  and  Mariusz URBAŃSKI
February 8, 2012. Published in Discrete and Continuous Dynamical Systems Ser. A, 32 (2012), no. 7, 2591–2606.
Abstract.

We estimate the Bowen parameters and the Hausdorff dimensions of the Julia sets of expanding finitely generated rational semigroups. We show that the Bowen parameter is larger than or equal to the ratio of the entropy of the skew product map and the Lyapunov exponent of with respect to the maximal entropy measure for . Moreover, we show that the equality holds if and only if the generators are simultaneously conjugate to the form by a Möbius transformation. Furthermore, we show that there are plenty of expanding finitely generated rational semigroups such that the Bowen parameter is strictly larger than .

Key words and phrases:
Complex dynamical systems, rational semigroups, expanding semigroups, Julia set, Hausdorff dimension, Bowen parameter, random complex dynamics
The first author thanks University of North Texas for support and kind hospitality. Research of the first author was partially supported by JSPS KAKENHI 21540216. Research of the second author supported in part by the NSF Grant DMS 0400481.
 
Hiroki Sumi
Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan
E-mail: sumi@math.sci.osaka-u.ac.jp
Web: http://www.math.sci.osaka-u.ac.jp/sumi/
 
Mariusz Urbański
Department of Mathematics, University of North Texas, Denton, TX 76203-1430, USA
E-mail: urbanski@unt.edu
Web: http://www.math.unt.edu/urbanski/

Mathematics Subject Classification (2001). Primary 37F35; Secondary 37F15.

1. Introduction

A rational semigroup is a semigroup generated by a family of non-constant rational maps , where denotes the Riemann sphere, with the semigroup operation being functional composition. A polynomial semigroup is a semigroup generated by a family of non-constant polynomial maps on The work on the dynamics of rational semigroups was initiated by A. Hinkkanen and G. J. Martin ([7]), who were interested in the role of the dynamics of polynomial semigroups while studying various one-complex-dimensional moduli spaces for discrete groups of Möbius transformations, and by F. Ren’s group ([35]), who studied such semigroups from the perspective of random dynamical systems.

The theory of the dynamics of rational semigroups on has developed in many directions since the 1990s ([7, 35, 15, 17, 18, 19, 20, 21, 22, 23, 30, 32, 25, 27, 16, 28, 29]). Since the Julia set of a rational semigroup generated by finitely many elements has backward self-similarity i.e.

(1.1)

(see [17, 19]), it can be viewed as a significant generalization and extension of both the theory of iteration of rational maps (see [11]) and conformal iterated function systems (see [10]). Indeed, because of (1.1), the analysis of the Julia sets of rational semigroups somewhat resembles “backward iterated functions systems”, however since each map is not in general injective (critical points), some qualitatively different extra effort in the cases of semigroups is needed. The theory of the dynamics of rational semigroups borrows and develops tools from both of these theories. It has also developed its own unique methods, notably the skew product approach (see [19, 20, 21, 22, 25, 26, 27, 29, 30, 31, 32]).

The theory of the dynamics of rational semigroups is intimately related to that of the random dynamics of rational maps. For the study of random complex dynamics, the reader may consult [5, 3, 4, 2, 1, 6, 29]. The deep relation between these fields (rational semigroups, random complex dynamics, and (backward) IFS) is explained in detail in the subsequent papers ([23, 25, 26, 27, 24, 28, 29]) of the first author.

In this paper, we deal at length with Bowen’s parameter (the unique zero of the pressure function) of expanding finitely generated rational semigroups (see Definition 2.12). In the usual iteration dynamics of a single expanding rational map, it is well known that the Hausdorff dimension of the Julia set is equal to the Bowen’s parameter. For a general expanding finitely generated rational semigroup , it was shown that the Bowen’s parameter is larger than or equal to the Hausdorff dimension of the Julia set ([18, 21]). If we assume further that the semigroup satisfies the “open set condition” (see Definition 3.2), then it was shown that they are equal ([21]). However, if we do not assume the open set condition, then there are a lot of examples such that the Bowen’s parameter is strictly larger than the Hausdorff dimension of the Julia set. In fact, the Bowen’s parameter can be strictly larger than two. Thus, it is very natural to ask when we have this situation and what happens if we have such a case. We will show the following.

Theorem 1.1 (Theorem 3.1).

For an expanding rational semigroup , the Bowen’s parameter satisfies

(1.2)

where denotes the skew product map associated with the multi-map (see section 2), and denotes the unique maximal entropy measure for (see [12, 19]). Moreover, the equality in the (1.2) holds if and only if we have a very special condition, i.e., there exists a Möbius transformation and a positive integer such that for each , is of the form .

Note that is equal to the entropy of The above result (Theorem 3.1) generalizes a weak form of A. Zdunik’s theorem ([34]), which is a result for the usual iteration of a single rational map. In fact, in the proof of the main result of our paper, Zdunik’s theorem is one of the key ingredients. We emphasize that in the main result of our paper, we can take the Möbius map which does not depend on

If each is a polynomial with , then by using potential theory, we can calculate in (1.2) in terms of and an integral related to fiberwise Green’s functions (see Lemmas 3.13, 3.14). From this calculation, we can prove the following.

Theorem 1.2 (Theorem 3.17).

Let and for each , let be a polynomial with If is an expanding polynomial semigroup, the postcritical set of in is bounded, where and , and , then there exists a Möbius transformation such that for each , is of the form

Thus, if the postcritical set of in is bounded and , then typically we have that Note that in the usual iteration dynamics of a single rational map, we always have

Therefore, we can say that there are plenty of expanding finitely generated polynomial semigroups for which the Bowen’s parameter is strictly larger than 2.

Note that combining these estimates of Bowen’s parameter and the “transversal family” type arguments, we will show that we have a large amount of expanding -generator polynomial semigroups such that the Julia set of has positive 2-dimensional Lebesgue measure ([33]).

We remark that, as illustrated in [24, 29], estimating the Hausdorff dimension of the Julia sets of rational semigroups plays an important role when we investigate random complex dynamics and its associated Markov process on For more details, see Remark 4.5 and [24, 29].

2. Preliminaries

In this section we introduce notation and basic definitions. Throughout the paper, we frequently follow the notation from [19] and [21].

Definition 2.1 ([7, 35]).

A “rational semigroup” is a semigroup generated by a family of non-constant rational maps , where denotes the Riemann sphere, with the semigroup operation being functional composition. A “polynomial semigroup” is a semigroup generated by a family of non-constant polynomial maps on For a rational semigroup , we set

and we call the Fatou set of . Its complement,

is called the Julia set of If is generated by a family , then we write

For the papers dealing with dynamics of rational semigroups, see for example [7, 35, 15, 17, 18, 19, 20, 21, 22, 23, 30, 32, 25, 26, 27, 16, 28, 29, 24], etc.

We denote by Rat the set of all non-constant rational maps on endowed with the topology induced by uniform convergence on Note that Rat has countably many connected components. In addition, each connected component of Rat is an open subset of Rat and has a structure of a finite dimensional complex manifold. Similarly, we denote by the set of all polynomial maps with endowed with the relative topology from Rat. Note that has countably many connected components. In addition, each connected component of is an open subset of and has a structure of a finite dimensional complex manifold.

Let be an open subset of and let For a holomorphic map , we denote by the norm of the derivative of at with respect to the spherical metric on

Definition 2.2.

For each , let be the space of one-sided sequences of -symbols endowed with the product topology. This is a compact metrizable space. For each , we define a map

by the formula

where and denotes the shift map. The transformation is called the skew product map associated with the multi-map We denote by the projection onto and by the projection onto . That is, and For each and , we put

Moreover, we denote by the norm of the derivative of at with respect to the spherical metric on We define

for each and we set

where the closure is taken with respect to the product topology on the space is called the Julia set of the skew product map In addition, we set and

Remark 2.3.

By definition, the set is compact. Furthermore, if we set , then, by [19, Proposition 3.2], the following hold:

  1. is completely invariant under ;

  2. is an open map on ;

  3. if and is contained in , then the dynamical system is topologically exact;

  4. is equal to the closure of the set of repelling periodic points of if , where we say that a periodic point of with period is repelling if .

Definition 2.4 ([21]).

A finitely generated rational semigroup is said to be expanding provided that and the skew product map associated with is expanding along fibers of the Julia set , meaning that there exist and such that for all ,

(2.1)
Definition 2.5.

Let be a rational semigroup. We put

and we call the postcritical set of . A rational semigroup is said to be hyperbolic if

Definition 2.6.

Let be a polynomial semigroup. We set We say that is postcritically bounded if is bounded in

Remark 2.7.

Let be a rational semigroup such that there exists an element with and such that each Möbius transformation in is loxodromic. Then, it was proved in [18] that is expanding if and only if is hyperbolic.

Definition 2.8.

We define

We also set (disjoint union). For every let be the length of For each and each , we put

Then we have the following.

Lemma 2.9 ([17, 31]).

is an open subset of (Rat

Definition 2.10.

We set

Lemma 2.11 ([27, 29]).

is open in

Definition 2.12.

Let and let be the skew product map associated with For each , let be the topological pressure of the potential with respect to the map (For the definition of the topological pressure, see [12].) We denote by the unique zero of (Note that the existence and the uniqueness of the zero of was shown in [21].) The number is called the Bowen parameter of the semigroup

We have the following fact, which is one of the main results of [31].

Theorem 2.13 ([31]).

The function is real-analytic and plurisubharmonic.

Definition 2.14.

For a subset of , we denote by the Hausdorff dimension of with respect to the spherical metric. For a Riemann surface , we denote by the set of all holomorphic isomorphisms of For a compact metric space , we denote by the space of all continuous complex-valued functions on , endowed with the supremum norm.

3. Results

In this section, we prove our main results. Note that for any , by Remark 2.3, [12], and [19], there exists a unique maximal entropy measure for and We start with the following.

Theorem 3.1.

Let Let and let for each Let be the maximal entropy measure for . Then the following statements (1) and (2) hold.

  • Suppose If

    then, the following items (a),(b),(c) hold.

    • We set

    • There exist an automorphism and complex numbers with such that for each ,

Proof.

We have that is convex and real-analytic ([21], [31]). Also,

From the convexity of , we obtain that

We now assume that and

Because of the convexity of again, we infer that

for all Let be the unique -conformal measure on for (see [21]). Let

be the operator, called the transfer operator, defined by the following formula

In virtue of [21], the limit exists, where denotes the constant function taking its only value Let Then

Thus

Since

(see [21]), it follows that By the uniqueness of maximal entropy measure of , we obtain that

(3.1)

Let be the operator defined as follows

Since , (3.1) implies that Thus, for any open subset of such that is injective, if is a Borel subset of , then Moreover, we have

Thus for -a.e. Since supp (see [21]), it follows that

(3.2)

Hence

(3.3)

Therefore, for each there exists a continuous function such that

(3.4)

Thus, for each -invariant Borel probability measure on , we have

Let be the topological pressure of with respect to the potential function It follows that for each with ,

(3.5)

In particular, is linear. Hence,

where denotes the maximal entropy measure for Therefore, by Zdunik’s theorem ([34]), it follows that for each with , there exists an and an element such that

(3.6)

In particular, there exists an element such that for each Suppose that there exists a such that If , then since each point of is a critical point of and , it contradicts (3.6). If , then since each point of is a critical point of and , it contradicts (3.6) again. Therefore, for each ,

(3.7)

for some Since is expanding and , it follows that for each By (3.5) and (3.7), it follows that for each ,

Therefore, Thus, we have completed the proof. ∎

Regarding Theorem 3.1, we give several remarks. In order to relate the Bowen parameter to the geometry of the Julia set we need the concept of the open set condition. We define it now.

Definition 3.2.

Let and let . Let also be a non-empty open set in We say that (or ) satisfies the open set condition (with ) if

for each with There is also a stronger condition. Namely, we say that (or ) satisfies the separating open set condition (with ) if

for each with

We remark that the above concept of “open set condition” (for “backward IFS’s”) is an analogue of the usual open set condition in the theory of IFS’s.

We introduce two other analytic invariants.

Definition 3.3 ([21]).

Let be a countable rational semigroup. For any and , we set

counting multiplicities. We also set

(if no exists with , then we set ). Furthermore, we put

The number is called the critical exponent of the Poincaré series of

Definition 3.4 ([21]).

Let , , and We put

counting multiplicities. Moreover, we set

(if no exists with , then we set ). Furthermore, we set

The number is called the critical exponent of the Poincaré series of

Remark 3.5.

Let , , and let Then, and Note that for almost every with respect to the Lebesgue measure, is a free semigroup and so we have and

Lemma 3.6 ([31]).

Let Then

Definition 3.7.

Let be a rational semigroup. Then, we define

Let us record the following fact proved in [21] .

Theorem 3.8 ([21]).

Let and let Then, by [21] and Lemma 3.6, we have for each If in addition to the above assumption, satisfies the open set condition, then

for each

In order to prove our second main theorem (see Theorem 3.17), we need some notation and lemmas from [29]. We shall provide the full proofs of these lemmas for the sake of completeness of our exposition and convenience of the readers.

Definition 3.9.

For each , we set

Definition 3.10 ([14, 8, 9, 29]).

Let Let be the skew product map associated with For any , we set

For any , let

where for each By the arguments in [14], for each , the limit exists, the function is subharmonic on , and is equal to the Green’s function on with pole at . Moreover, is continuous on Let , where Note that by the argument in [8, 9], is a Borel probability measure on such that Furthermore, for each , let , where runs over all critical points of in , counting multiplicities.

Remark 3.11 ([19]).

Let Let be the skew product map associated with Also, let and let be the Bernoulli measure on with respect to the weight Suppose that for each Then, there exists a unique -invariant Borel probability ergodic measure on such that and

where denotes the relative metric entropy of with respect to , and denotes the space of ergodic measures for (see [19]). The measure is called the maximal relative entropy measure for with respect to

Lemma 3.12 ([29]).

Let and let Let Let be the skew product associated with Let be the Bernoulli measure on with respect to the weight Let be a Borel probability measure on defined by

for any continuous function on , where is the measure coming from Definition 3.10. Then, is an -invariant ergodic measure, , and is the maximal relative entropy measure for with respect to (see Remark 3.11).

Proof.

By the argument of the proof of [9, Theorem 4.2(i)], is -invariant and ergodic, and Moreover, the argument of the proof of [9, Theorem 5.2(i)], yields that

Combining this with [19, Theorem 1.3(e)(f)], it follows that is the unique maximal relative entropy measure for with respect to

Lemma 3.13 ([29]).

Let . Let