Bowen Parameter and Hausdorff Dimension for Expanding Rational Semigroups

We consider the dynamics of rational semigroups (semigroups of rational maps) on the Riemann sphere. We estimate the Bowen parameters (zeros of the pressure functions) 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 $F$ and the Lyapunov exponent of $F$ with respect to the maximal entropy measure for $F$. Moreover, we show that the equality holds if and only if the generators $f_{j}$ are simultaneously conjugate to the form $a_{j}z^{\pm d}$ by a linear fractional transformation. Furthermore, we show that there are plenty of expanding finitely generated rational semigroups such that the Bowen parameter is strictly larger than two.


Introduction
A rational semigroup is a semigroup generated by a family of non-constant rational maps g :Ĉ →Ĉ, whereĈ 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 onĈ. 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.
(1.1) J(G) = f −1 1 (J(G)) ∪ · · · ∪ f −1 s (J(G)), (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 f j 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 δ (the unique zero of the pressure function) of expanding finitely generated rational semigroups f 1 , . . . , f s (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 f 1 , . . . , f s , 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 G = f 1 , . . . , f m , the Bowen's parameter δ satisfies wheref denotes the skew product map associated with the multi-map f = (f 1 , . . . , f s ) (see section 2), and µ denotes the unique maximal entropy measure forf (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 ϕ and a positive integer d 0 such that for each j, ϕf j ϕ −1 (z) is of the form a j z ±d 0 .
Note that log( s j=1 deg(f j )) is equal to the entropy off . 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 ϕ which does not depend on j.
If each f j is a polynomial with deg(f j ) ≥ 2, then by using potential theory, we can calculate log f ′ dµ in (1.2) in terms of deg(f j ) 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 s ∈ N and for each j = 1, . . . , s, let f j be a polynomial with deg(f j ) ≥ 2. If G = f 1 , . . . , f s is an expanding polynomial semigroup, the postcritical j=1 d j , and δ ≤ 2, then there exists a Möbius transformation ϕ such that for each j, ϕf j ϕ −1 (z) is of the form a j z s .
Thus, if the postcritical set of G in C is bounded and (log d)/( s j=1 d j d log d j ) ≥ 2, then typically we have that δ > 2. Note that in the usual iteration dynamics of a single rational map, we always have δ ≤ 2.
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 2-generator polynomial semigroups G such that the Julia set of G 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 onĈ. For more details, see Remark 4.5 and [24,29].

Preliminaries
In this section we introduce notation and basic definitions. Throughout the paper, we frequently follow the notation from [19] and [21]. 7,35]). A "rational semigroup" G is a semigroup generated by a family of non-constant rational maps g :Ĉ →Ĉ, whereĈ 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 onĈ. For a rational semigroup G, we set F (G) := {z ∈Ĉ | G is normal in a neighborhood of z} and we call F (G) the Fatou set of G. Its complement, is called the Julia set of G. If G is generated by a family {f i } i , then we write G = f 1 , f 2 , . . . .
We denote by Rat the set of all non-constant rational maps onĈ endowed with the topology induced by uniform convergence onĈ. Note that Rat has countably many connected components. In addition, each connected component U of Rat is an open subset of Rat and U has a structure of a finite dimensional complex manifold. Similarly, we denote by P the set of all polynomial maps g :Ĉ →Ĉ with deg(g) ≥ 2 endowed with the relative topology from Rat. Note that P has countably many connected components. In addition, each connected component U of P is an open subset of P and U has a structure of a finite dimensional complex manifold.

Definition 2.4 ([21]).
A finitely generated rational semigroup G = f 1 , . . . , f s is said to be expanding provided that J(G) = ∅ and the skew product mapf : Σ s ×Ĉ → Σ s ×Ĉ associated with f = (f 1 , . . . , f s ) is expanding along fibers of the Julia set J(f ), meaning that there exist η > 1 and C ∈ (0, 1] such that for all n ≥ 1, where · denotes the absolute value of the spherical derivative.
Definition 2.5. Let G be a rational semigroup. We put . . , f s be a rational semigroup such that there exists an element g ∈ G with deg(g) ≥ 2 and such that each Möbius transformation in G is loxodromic. Then, it was proved in [18] that G is expanding if and only if G is hyperbolic.
Then we have the following.
(For the definition of the topological pressure, see [12].) We denote by δ(f ) the unique zero of t → P (t, f ). (Note that the existence and the uniqueness of the zero of P (t, f ) was shown in [21].) The number δ(f ) is called the Bowen parameter of the semigroup f = (f 1 , . . . , f s ) ∈ Exp(s).
We have the following fact, which is one of the main results of [31]. Definition 2.14. For a subset A ofĈ, we denote by HD(A) the Hausdorff dimension of A with respect to the spherical metric. For a Riemann surface S, we denote by Aut(S) the set of all holomorphic isomorphisms of S. For a compact metric space X, we denote by C(X) the space of all continuous complex-valued functions on X, endowed with the supremum norm.
(b) There exist an automorphism ϕ ∈ Aut(Ĉ) and complex numbers a 1 , . . . , a s with a 1 = 1 such that for each j = 1, . . . , s, [31]). Also, From the convexity of P (t, f ), we obtain that We now assume that d 1 ≥ 2 and Because of the convexity of P (t, f ) again, we infer that [21]). Let be the operator, called the transfer operator, defined by the following formula In virtue of [21], the limit α := lim l→∞ L l ν (1) ∈ C(J(f )) exists, where 1 denotes the constant function taking its only value 1. Let τ := αν. Then Thus Since (see [21]), it follows that h τ (f ) = log d. By the uniqueness of maximal entropy measure of (f , J(f )), we obtain that Let L µ : C(J(f )) → C(J(f )) be the operator defined as follows Since L * µ (µ) = µ,  [21]), it follows that

Moreover, we have
Therefore, for each w ∈ Σ * s there exists a continuous function α w : Thus, for each f w -invariant Borel probability measure β on J(f w ), we have Let p(t, w) be the topological pressure of f w : J(f w ) → J(f w ) with respect to the potential function −t log f ′ w . It follows that for each w ∈ Σ * s with deg(f w ) ≥ 2, In particular, t → p(t, w) is linear. Hence, where µ w denotes the maximal entropy measure for f w : J(f w ) → J(f w ). Therefore, by Zdunik's theorem ( [34]), it follows that for each w ∈ Σ * s with deg(f w ) ≥ 2, there exists an n w ∈ Z \ {0, ±1} and an element ψ w ∈ Aut(Ĉ) such that (3.6) ψ w • f w • ψ −1 w (z) = z nw for every z ∈Ĉ. In particular, there exists an element ϕ ∈ Aut(Ĉ) such that ϕ • f 1 • ϕ −1 (z) = z ±d 1 for each z ∈Ĉ. Suppose that there exists a j ∈ {1, . . . , s} such that (ϕ and ♯A ≥ 3, it contradicts (3.6) again. Therefore, for each j ∈ {1, . . . , s}, for some a j ∈ C \ {0}. Since G is expanding and d 1 ≥ 2, it follows that d j ≥ 2 for each j = 1, . . . , s. By (3.5) and (3.7), it follows that for each j, Therefore, d 1 = · · · = d s . 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.
There is also a stronger condition. Namely, we say that f (or G) satisfies the separating open set condition (with U) if 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.    Let us record the following fact proved in [21] . . . , f s . Then, by [21] and Lemma 3.6, we have HD( . If in addition to the above assumption, f satisfies the open set condition, then for each z ∈Ĉ \ (A(G) ∪ P (G)).
In particular, if, in addition to the assumptions of our lemma, f 1 , . . . , f m is postcritically bounded, then Proof. Let Let τ be the Bernoulli measure on Σ s with respect to the weight p. By [19], µ is equal to the maximal relative entropy measure forf with respect to (σ, τ ). By Lemma 3.13, the statement of our lemma holds.
We now give a lower estimate of the Hausdorff dimension of Julia sets of expanding finitely generated polynomial semigroups satisfying the open set condition. Let µ be the maximal entropy measure forf : Σ s ×Ĉ → Σ s ×Ĉ (see [19]). Let τ be the Bernoulli measure on Σ s with respect to the weight ( d 1 d , . . . , ds d ).
(b) There exists an element ϕ ∈ Aut(C) and complex numbers a 1 , . . . , a s with a 1 = 1 such that for each j = 1, . . . , s, ϕf If, in addition to the assumptions of our lemma, f ∈ Epb(s), then Proof. By Theorem 3.1, Lemma 3.14 and Theorem 3.8, we obtain the statement of our Theorem.

Remarks and examples
In this section we collect some remarks and construct relevant examples illustrating our main theorems.
varies only inside the Julia set of the polynomial semigroup generated by Γ, and under some condition, the function T ∞ :Ĉ → [0, 1] is continuous inĈ. If the Hausdorff dimension of the Julia set is strictly less than two, then it means that T ∞ :Ĉ → [0, 1] is a complex version of devil's staircase (Cantor function) ( [23,29]). For example, setting g 1 (z) := z 2 − 1, g 2 (z) := z 2 4 , f 1 := g 2 1 , and f 2 := g 2 2 , we consider the random dynamics onĈ such that at every step we choose a map f j with probability 0 < p j < 1, where p 1 + p 2 = 1. Then the function T ∞ of probability of tending to ∞ is continuous onĈ and varies exactly on the Julia set ( Figure 1) of the polynomial semigroup f 1 , f 2 , whose Hausdorff dimension is strictly less than two (see [23,29]).