Semi-hyperbolic fibered rational maps and rational semigroups ^{∗}

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

http://www.math.sci.osaka-u.ac.jp/∼sumi/welcomeou-e.html

June 10, 2006

Abstract

This paper is based on the author’s previous work [S4]. We consider fiber-preserving complex dynamics on fiber bundles whose fibers are Riemann spheres and whose base spaces are compact metric spaces. In this context, without any assumption on (semi-)hyperbolicity, we show that the fiberwise Julia sets are uniformly perfect. From this result, we show that, for any semigroup G generated by a compact family of rational maps on the Riemann sphere \overline{ℂ} of degree two or greater, the Julia set of any subsemigroup of G is uniformly perfect.

We define the semi-hyperbolicity of dynamics on fiber bundles and show that, if the dynamics on a fiber bundle is semi-hyperbolic, then the fiberwise Julia sets are porous, and the dynamics is weakly rigid. Moreover, we show that if the dynamics is semi-hyperbolic and the fiberwise maps are polynomials, then under some conditions, the fiberwise basins of infinity are John domains. We also show that the Julia set of a rational semigroup (a semigroup generated by rational maps on \overline{ℂ}) that is semi-hyperbolic, except at perhaps finitely many points in the Julia set, and which satisfies the open set condition, is either porous or equal to the closure of the open set. Furthermore, we derive an upper estimate of the Hausdorff dimension of the Julia set.

1 Introduction

To investigate random one-dimensional complex dynamics, the dynamics of semigroups generated by rational maps on the Riemann sphere \overline{ℂ} and fiberpreserving holomorphic dynamics on fiber bundles that appear in complex dynamics in several dimensions, we consider the dynamics of fibered rational maps; that is, we consider fiber-preserving complex dynamical systems on fiber bundles whose fibers are Riemann spheres and whose base spaces are general compact metric spaces. The notion of the dynamics of fibered rational maps, which is a generalized notion of the “dynamics of the fibered polynomial maps” presented by O. Sester([Se1], [Se2], [Se3]), was introduced by M. Jonsson in [J2]. Researches on the dynamics of semigroups generated by rational maps on the Riemann sphere ([HM1], [HM2], [HM3] [GR], [Bo], [St1], [St2], [St3], [SSS], [S1], [S2], [S3], [S4], [S5], [S6], [S7]), the random iterations of rational functions([FS], [BBR], [Br]) and polynomial skew products on ℂ^{2} ([H1], [H2], [J1]) are directly related to this subject. For polynomial skew products (the dynamics of fibered polynomials) whose base spaces are general compact metric spaces, see O.Sester’s work [Se1], [Se2] and [Se3]. In [Se3], he investigated the quadratic case in detail. In particular, he developed a combinatorial theory for quadratic fibered polynomials and constructed an abstract space of combinatorics. Moreover, he demonstrated the realizability and rigidity of abstract combinatorics.

∗published in Ergodic Theory and Dynamical Systems (2006), 26, 893-922.

For the ergodic theory of random diffeomorphisms, see [K].

For the Hausdorff dimension of the Julia set of a (semi-) hyperbolic rational semigroup, see [S4] and [S7].

For dynamics of semi-hyperbolic transcendental semigroups, see [KS1], [KS2]. In this paper, by applying some of the results in [S4] obtained by the au-

thor, we show that semi-hyperbolicity along the fibers of a fibered rational map implies that the fiberwise Julia sets are k-porous, where the constant k does not depend on any points of the base space (hence the upper box dimensions of the fiberwise Julia sets are uniformly less than a constant that is strictly less than 2) (Theorem 1.16); that the dynamics have a kind of weak rigidity (Theorem 1.19); and that if the fiberwise maps are polynomials with some conditions, then the fiberwise basins of infinity are c-John domains, where c is a constant not depending on any points in the base spaces(Theorem 1.12). This generalizes a result from [CJY] and a result from [Se3]. These results are stated in Section 1.1, and the proofs are given in Section 2.5. For research on the semi-hyperbolicity of the usual dynamics of rational functions, see [CJY] and [Ma].

In Section 1.2, we present several results on the dynamics of rational semigroups; i.e., on the semigroups generated by rational maps on the Riemann sphere. We show that the Julia set of a finitely generated rational semigroup that is semi-hyperbolic, except at most finitely many points in the Julia set, and that satisfies the “(backward) open set condition” (which is a generalized notion in research on self-similar sets constructed by some similarity transformations) with an open set, is either porous (and hence the upper box dimension is strictly less than 2) or equal to the closure of the open set (Theorem 1.25). Furthermore, under the same assumptions, we show that the Hausdorff dimension of the Julia set is less than the infimum of numbers δ, which allows the construction of a “δ-subconformal measure” that is less than the critical exponent of the semigroup (Theorem 1.28). The proofs of these results are given in Section 2.6.

A proof of the above results requires the potential theoretic stories (Section 2.3), distortion lemmas for holomorphic proper maps, various results on non-constant limit functions, and the continuity of the fiberwise Julia sets with respect to points in the base spaces (this is an important property and is not easy to show), from [S4]. Note that the upper semicontinuity of fiberwise Julia sets with respect to points in the base spaces does not hold in general.

Furthermore, in Section 1.1, we show that the fiberwise Julia sets of a fibered rational map are C_{1}-uniformly perfect and that the Hausdorff dimensions are greater than a positive constant C_{2}, where C_{1} and C_{2} are constants that do not depend on any points in the base space, if the fiberwise maps are of degree two or greater, even if semi-hyperbolicity is not assumed (Theorem 1.6). The proof is given in Section 2.5. This result on uniform perfectness is used to show the result on Johnness. Furthermore, from the uniform perfectness of fiberwise Julia sets, we show that, for any semigroup G generated by a compact family of rational maps on \overline{ℂ} of degree two or greater, there exists a positive constant C such that the Julia set of any subsemigroup of G is C-uniformly perfect (Theorem 1.21). The proof, which is given in Section 2.6, is based on combining the density of repelling fixed points in the Julia set (an application of Ahlfors’s five-island theorem) with the potential theory in Section 2.3.

1.1 Results on fibered rational maps

In this section, we state the results on fibered rational maps. The proofs are given in Section 2.5. First, we provide some notation and definitions regarding the dynamics of fibered rational maps.

Definition 1.1. ( [J2]) A triplet (π, Y, X) is called a “\overline{ℂ}-bundle” if

1. Y and X are compact metric spaces,

2. π : Y → X is a continuous and surjective map,

3. and there exists an open covering {U_{i}} of X such that, for each i, there exists a homeomorphism Φ_{i} : U_{i} × \overline{ℂ} → π^{−1}(U_{i}) such that Φ_{i}(\{x\} × \overline{ℂ}) = π^{−1}\{x\} and Φ_{j}^{−1} ∘ Φ_{i} : \{x\} × \overline{ℂ} → \{x\} × \overline{ℂ} is a Möbius map for each x ∈ U_{i} ∩ U_{j}, under the identification \{x\} × \overline{ℂ} ≅ \overline{ℂ}.

Remark 1. By condition 3, each fiber Y_{x} := π^{−1}{x} has a complex structure. Furthermore, given x_{0} ∈ X, one may find a continuous family i_{x} : \overline{ℂ} → Y_{x} of homeomorphisms, for x close to x_{0}. Such a family {i_{x}} is called a “local parameterization”. Since X is compact, we may assume throughout this paper that there exists a compact subset M_{0} of the set of Möbius transformations of \overline{ℂ} such that i_{x} ∘ j_{x}^{−1} ∈ M_{0} for any two local parameterizations \{i_{x}\} and \{j_{x}\} .

Moreover, throughout this paper, we assume the following condition:

• there exists a smooth (1, 1)-form ω_{x} > 0 inducing the distance on Y_{x} from Y, and x ↦ ω_{x} is continuous. That is, if {i_{x}} is a local parameterization, then the pull back i_{x}^{∗}ω_{x} is a positive smooth form on \overline{ℂ} that depends continuously on x with respect to the C^{∞} topology.

Notation: If (π, Y = X × \overline{ℂ}, X) is a trivial \overline{ℂ}-bundle, then we denote by π\overline{ℂ} : Y → \overline{ℂ} the second projection.

Definition 1.2. Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y and g : X → X be continuous maps. Let f be called a fibered rational map over g (or a rational map fibered over g), if

1. π ∘ f = g ∘ π 2. f|_{Y_{x}} : Y_{x} → Y_{g(x)} is a rational map, for any x ∈ X. That is, (i_{g_{x}})^{−1} ∘ f ∘ i_{x}

is a rational map from \overline{ℂ} to itself, for any local parameterization i_{x} at x ∈ X and i_{g(x)} at g(x).

Notation: If f : Y → Y is a fibered rational map over g : X → X, then we set f_{x}^{n} = f^{n}|_{Y_{x}} : Y_{x} → Y_{g^{n}(x)}, for any x ∈ X and n ∈ ℕ. Moreover, we set f_{x} = f|_{Y_{x}} : Y_{x} → Y_{g(x)}. Furthermore, we set d_{n}(x) = deg(f_{x}^{n}) and d(x) = d_{1}(x), for any x ∈ X and n ∈ ℕ. Note that the definition of fibered rational map implies that the function x ↦ d(x) is continuous on X.

Definition 1.3. Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y be a fibered rational map over g : X → X. Then, for any x ∈ X, we denote by F_{x}(f)(or simply F_{x}) the set of points y ∈ Y_{x} that has a neighborhood U of y in Y_{x} such that {f_{x}^{n}}_{n∈ℕ} is a normal family in U; that is, y ∈ F_{x} if and only if the family Q_{x}^{n} = i_{x_{n}}^{−1} ∘ f_{x}^{n} ∘ i_{x} of rational maps on \overline{ℂ} (x_{n} := g^{n}(x) ) is normal near i_{x}^{−1}(y). Note that, by Remark 1, this does not depend on the choices of local parameterizations at x and x_{n}. Equivalently, F_{x} is the open subset of Y_{x}, where the family {f_{x}^{n}} of mappings from Y_{x} into Y is locally equicontinuous. We set J_{x}(f)(or simply J_{x})= Y_{x} \ F_{x}.

Furthermore, we set J̃(f) = \overline{⋃_{x∈X} J_{x}}, where the closure is taken in the space Y, F̃(f) = Y \ J̃(f), and Ĵ_{x}(f)(or simply Ĵ_{x})= J̃(f) ∩ Y_{x}, for each x ∈ X.

Remark 2. There exists a fibered rational map f : Y → Y such that ⋃_{x∈X} J_{x} is NOT compact.

Definition 1.4. ([S4]) Let h_{1}, …, h_{m} be non-constant rational maps. Let Σ_{m} = {1, …, m}^{ℕ} be the space of one-sided infinite sequences of m symbols, and σ : Σ_{m} → Σ_{m} be the shift map; that is, σ is defined by σ((w_{1}, w_{2}, …)) = (w_{2}, w_{3}, …). Let X be a compact subset of Σ_{m} such that σ(X) ⊂ X. Let Y = X × \overline{ℂ} and π : Y → X be the natural projection. Then (π, Y, X) is a \overline{ℂ}- bundle. Let f : Y → Y be a map defined by: f((w, y)) = (σ(w), h_{w_{1}}(y)), where w = (w_{1}, w_{2}, …). Then, f : Y → Y is a fibered rational map over σ : X → X.

In the above, if X = Σ_{m}, then f : Y → Y is called the fibered rational map associated with the generator system {h_{1}, … h_{m}} (of the rational semigroup G = ⟨h_{1}, …, h_{m}⟩). (See Section 1.2 for the definition of rational semigroups.)

Now we present a result on uniform perfectness.

Definition 1.5. Let C be a positive number. Let K be a closed subset of \overline{ℂ}. We say that K is C-uniformly perfect if, ♯K ≥ 2 and for any doubly connected domain A in \overline{ℂ} such that A separates K; i.e., such that A ⊂ \overline{ℂ} \ K and both of the two connected components of \overline{ℂ} \ A have non-empty intersections with K, mod A (the modulus of A; for the definition, see [LV]) is less than C.

Theorem 1.6. (Uniform perfectness) Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y be a fibered rational map over g : X → X such that d(x) ≥ 2, for any x ∈ X. (In this theorem, we do NOT assume semi-hyperbolicity.) Then, we have the following.

1. There exists a positive constant C such that, for any x ∈ X, J_{x} and Ĵ_{x} are C-uniformly perfect. Furthermore, there exists a positive constant C_{1} such that for each x ∈ X, we have diam J_{x} ≥ C_{1}, with respect to the distance in Y_{x} induced by ω_{x}. Moreover, there exists a positive constant C_{2} such that, for each x ∈ X, the Hausdorff dimension dim_{H}(J_{x}) of J_{x}, with respect to the distance on Y_{x} induced by ω_{x}, satisfies the condition that dim_{H}(J_{x}) ≥ C_{2}. (Note that C_{1} and C_{2} do not depend on x ∈ X. )

2. Suppose further that f(F̃(f)) ⊂ F̃(f)( for example, assume that g : X → X is an open map). If a point z ∈ Y satisfies f_{π(z)}^{n}(z) = z and (f_{π(z)}^{n})′(z) = 0 for some n ∈ ℕ and z ∈ Ĵ_{π(z)}, then z belongs to the interior of Ĵ_{π(z)} with respect to the topology of Y_{π(z)}.

Theorem 1.6 is now used to show a result on Johnness (Theorem 1.12) and Theorem 1.21.

Example 1.7. Let z_{0} ∈ \overline{ℂ} be a point. Let h_{1} and h_{2} be two rational maps on \overline{ℂ} of degree two or greater. Let f : Σ_{2} × \overline{ℂ} → Σ_{2} × \overline{ℂ} be the fibered rational map associated with the generator system {h_{1}, h_{2}}. Suppose that z_{0} is a superattracting fixed point of h_{1} and is a repelling fixed point of h_{2}. Then, it can easily be seen that z_{0} ∈ Ĵ_{x}, where x = (1, 1, …) ∈ Σ_{2}. Since the shift map σ : Σ_{2} → Σ_{2} is an open map, by Theorem 1.6 it follows that z_{0} belongs to the interior of Ĵ_{x}.

Remark 3. Theorem 1.6 generalizes Theorem 3.1 in [Br].

Remark 4. Uniform perfectness implies many useful properties ([BP],[Po],[Su]). This terminology was introduced in [Po]. For a survey on uniform perfectness, see [Su]. We now consider the following:

1. In [BP], it was shown that a closed subset K of \overline{ℂ} with ♯K ≥ 3 is Cuniformly perfect if and only if there exists a constant δ such that, for any component U of \overline{ℂ} \ K,

λ_{U}(z) > δ/dist(z, ∂U),  (1)

where λ_{U}(z) denotes the density of the hyperbolic metric of U at z and dist(z, ∂U) denotes the Euclidian distance of the point z from the set ∂U. (If K is bounded and U is the unbounded component of \overline{ℂ} \ K, then inequality (1) will hold only for all z ∈ U sufficiently close to K.) In the above discussion, δ depends only on C, and C depends only on δ. This fact will be used to show Theorem 1.12. Detailed inequalities regarding the relationships among δ, C and other invariants are presented in [Su].

2. If a closed subset K of \overline{ℂ} is C-uniformly perfect, then the Hausdorff dimension dim_{H}(K) of K with respect to the spherical metric satisfies dim_{H}(K) ≥ C′ > 0, where C′ is a positive constant that depends only on C (Theorem 7.2 in [Su]).

Notation :

• Let Z_{1} and Z_{2} be two topological spaces and g : Z_{1} → Z_{2} be a map. For any subset A of Z_{2}, We denote by c(A, g) the set of all connected components of g^{−1}(A).

• For any y ∈ \overline{ℂ} and δ > 0, we set B(y, δ) = \{y′ ∈ \overline{ℂ} | d(y, y′) < δ\}, where d is the spherical distance. Similarly, for any y ∈ ℂ and δ > 0, we set D(y, δ) = {y′ ∈ ℂ | |y − y′| < δ}.

• Let (π, Y, X) be a \overline{ℂ}-bundle. For any y ∈ Y and r > 0, we set B̃(y, r) = {y′ ∈ Y_{π(y)} | d_{π(y)}(y′, y) < r}, where, for each x ∈ X, we denote by d_{x} the distance on Y_{x} induced by the form ω_{x}.

We now define the semi-hyperbolicity of fibered rational maps.

Definition 1.8. (semi-hyperbolicity of fibered rational maps) Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y be a fibered rational map over g : X → X. Let N ∈ ℕ. We denote by SH_{N}(f) the set of points z ∈ Y such that there exists a positive number δ, a neighborhood U of π(z), and a local parameterization {i_{x}} in U such that, for any x ∈ U, any n ∈ ℕ, any x_{n} ∈ g^{−n}(x) and any V ∈ c(i_{x}(B(i_{π(z)}^{−1}(z), δ)), f_{x_{n}}^{n}), we have that

deg(f_{x_{n}}^{n} : V → i_{x}(B(i_{π(z)}^{−1}(z), δ))) ≤ N.

We set UH(f) = Y \ ⋃_{N∈ℕ} SH_{N}(f). A point z ∈ SH_{N}(f) is called a semihyperbolic point of degree N. Furthermore, we say that f is semi-hyperbolic (along fibers) if J̃(f) ⊂ ⋃_{N∈ℕ} SH_{N}(f). This is equivalent to J̃(f) ⊂ SH_{N}(f), for some N ∈ ℕ.

Definition 1.9. Let f : Y → Y be a rational map fibered over g : X → X. Then, we set C(f) := ⋃_{x∈X}{critical points of f_{x}} (⊂ Y). This is called the fiber critical set of fibered rational map f. Furthermore, we set P (f) = \overline{⋃_{n∈ℕ} ⋃_{x∈X} f_{x}^{n}( critical points of f_{x})} where the closure is taken in Y. This is called the fiber postcritical set of the fibered rational map f. Furthermore, we say that f : Y → Y is hyperbolic along fibers if P (f) ⊂ F (f). (Then f is semi-hyperbolic along fibers with the constant N = 1.)

We now present a result on the John condition. Notation: Let y ∈ ℂ and b ∈ \overline{ℂ} be two distinct points. Furthermore, let E be a curve in \overline{ℂ} joining y to b such that E \ \{b\} ⊂ ℂ. Then, for any c ≥ 1, we set

car (E, c, y, b) = ⋃_{z∈E\\{y,b\}} D(z, \frac{|y − z|}{c}).

This is called the c-carrot with core E and vertex y joining y to b.

Definition 1.10. Let V be a subdomain of \overline{ℂ} such that ∞ ∈ \overline{ℂ} \ ∂V. Let c ≥ 1 be a number. Then, we say that V is a c-John domain if there exists a point y_{0} ∈ \overline{V} such that, for any y ∈ V \ \{y_{0}\}, there exists a curve E joining y_{0} to y such that E \ {y_{0}} ⊂ ℂ and car (E, c, y, y_{0}) ⊂ V. In what precedes, the point y_{0} is called the center of the John domain V.

Definition 1.11. Let (π, Y = X × \overline{ℂ}, X) be a trivial \overline{ℂ}-bundle. A fibered rational map f : Y → Y over g : X → X is called a fibered polynomial map over g : X → X if f is of the form: f(x, y) = (g(x), q_{x}(y)) for (x, y) ∈ X × \overline{ℂ}, where q_{x} is a polynomial for each x ∈ X. For such an f, we set A_{x}(f) := {y ∈ Y_{x} | π\overline{ℂ}(f_{x}^{n}(y)) → ∞\}.

Theorem 1.12. (Johnness) Let (π, Y = X ×\overline{ℂ}, X) be a trivial \overline{ℂ}-bundle. Let f : Y → Y be a semi-hyperbolic fibered polynomial map over g : X → X such that d(x) ≥ 2, for any x ∈ X. Assume that either 1. J_{x}(f) is connected for each x ∈ X, or

2. the map x ↦ f_{x} from X to the space of polynomials is locally constant (here, we identify f_{x} with a polynomial on \overline{ℂ}) and inf\{d(a, b) | a ∈ π\overline{ℂ}(J_{x}), b ∈ π\overline{ℂ}(A_{x}(f) ∩ C(f)), x ∈ X\} > 0.

Then, there exists a positive constant c such that, for any x ∈ X, the basin of infinity A_{x}(f) is a c-John domain.

Remark 5. • For a fibered polynomial map f such that d(x) ≥ 2 for each x ∈ X, we have that J_{x}(f) is connected for each x ∈ X if and only if π\overline{ℂ}(P (f)) \ \{∞\} is bounded in ℂ.

• In Theorem 1.12, if X is a set consisting of one point, then f is semihyperbolic if and only if the basin of infinity is a John domain([CJY]). In [Se3], O. Sester made the same assertion as that presented in Theorem 1.12 for “non-recurrent quadratic fibered polynomials” with connected fiberwise Julia sets.

Remark 6. Johnness likewise implies many useful properties ([NV], [Jone]).For example, if V is a John domain, then the following facts hold.

• If ∞ ∈ V, then the center of V is ∞.

• Let a ∈ ∂V and b ∈ V. Then there exists a curve E joining a to b and a constant c such that car (E, c, a, b) ⊂ V. In particular, a is accessible from b.

• V is finitely connected at any point in ∂V: that is, if y ∈ ∂V, then there exists an arbitrarily small open neighborhood U of y in \overline{ℂ} such that U ∩ V has only finitely many connected components.

• If V is simply connected, then it follows that ∂V is locally connected.

• ∂V is holomorphically removable: that is, if φ : \overline{ℂ} → \overline{ℂ} is a homeomorphism and is holomorphic on \overline{ℂ} \ ∂V, then φ is holomorphic on \overline{ℂ}. From this fact, we deduce that the two-dimensional Lebesgue measure of ∂V is equal to zero.

In order to present results on porosity and other properties, let us first establish some technical conditions.

Definition 1.13 (Condition(C1)). Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y be a fibered rational map over g : X → X. We say that f satisfies condition (C1) if there exists a family {D_{x}}_{x∈X} of topological disks with D_{x} ⊂ Y_{x}, x ∈ X such that the following conditions are satisfied:

1. for each x ∈ X, there exists a point z_{x} ∈ Y_{x} and a positive number r_{x} such that D_{x} = B̃(z_{x}, r_{x}),

2. \overline{⋃_{x∈X} ⋃_{n≥0} f_{x}^{n}(D_{x})} ⊂ F̃(f),

3. for any x ∈ X, it holds that diam(f_{x}^{n}(D_{x})) → 0, as n → ∞, and

4. inf_{x∈X} r_{x} > 0.

Definition 1.14 (Condition(C2)). Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y be a fibered rational map over g : X → X. We say that f satisfies condition (C2) if, for each x_{0} ∈ X, there exists an open neighborhood O of x_{0} and a family {D_{x}}_{x∈O} of topological disks with D_{x} ⊂ Y_{x}, x ∈ O such that the following conditions are satisfied:

1. for each x ∈ O, there exists a point z_{x} ∈ Y_{x} and a positive number r_{x} such that D_{x} = B̃(z_{x}, r_{x}),

2. \overline{⋃_{x∈O} ⋃_{n≥0} f_{x}^{n}(D_{x})} ⊂ F̃(f),

3. for any x ∈ O, it holds that diam(f_{x}^{n}(D_{x})) → 0, as n → ∞, and

4. x ↦ D_{x} is continuous in O.

Remark 7. 1. Let (π, Y = X ×\overline{ℂ}, X) be a trivial \overline{ℂ}-bundle. Let f : Y → Y be a fibered polynomial map over g : X → X such that d(x) ≥ 2 for each x ∈ X. Then, setting D_{x} = D, where D is a small neighborhood of infinity, for each x ∈ X, the fibered rational map f satisfies condition (C2) with the family of disks (D_{x})_{x∈X}.

2. Let \{h_{1}, … h_{m}\} be non-constant rational functions on \overline{ℂ} with deg(h_{1}) ≥ 2. Let f : Y → Y be the fibered rational map associated with the generator system {h_{1}, …, h_{m}} of rational semigroup G = ⟨h_{1}, …, h_{m}⟩ (See Section 1.2). Suppose that f is semi-hyperbolic along fibers and that π\overline{ℂ}(J̃(f)) = J(G) (See Section 1.2 and [S5]) is not equal to the Riemann sphere. Moreover, suppose that each element of Aut (\overline{ℂ}) ∩ G ( if this is not empty ) is loxodromic. Then, it follows that f satisfies condition (C2). Actually, there exists an attracting fixed point a of some element of G in F (G). Since G is semi-hyperbolic, if D_{x} = D(a, ϵ), for each x ∈ Σ_{m} where ϵ is a positive number, then f satisfies condition (C2) with the family of disks (D_{x})_{x∈Σ_{m}}. For more details, see Theorem 1.35 and Remark 5 in [S4].

We now present a result on porosity.

Definition 1.15. Let (Y, d) be a metric space. Let k be a constant such that 0 < k < 1. Let J be a subset of Y. We say that J is k-porous if, for any x ∈ J and any positive number r, there exists a ball in {y ∈ Y | d(y, x) < r} \ J with radius at least kr.

Remark 8. If Y is the Euclidean space ℝ^{n} and d is the Euclidean metric, the upper box dimension of any k-porous bounded set J in ℝ^{n} is less than n−c(k, n), where c(k, n) is a positive constant that depends only on k and n (Proposition 4.2 and the proof in [PR]).

Theorem 1.16. (Porosity) Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y be a fibered rational map over g : X → X. Suppose that f satisfies condition (C1) and that f is semi-hyperbolic. Then there exists a constant 0 < k < 1 such that, for each x ∈ X, J_{x} is k-porous in Y_{x}. In particular, there exists a constant 0 ≤ c < 2 such that, for each x ∈ X, dim_{H}(J_{x}) ≤ \overline{dim}_{B}(J_{x}) ≤ c, where dim_{H} denotes the Hausdorff dimension and \overline{dim}_{B} denotes the upper box dimension with respect to the metric on Y_{x} induced by ω_{x} (ω_{x} is of the form in Remark 1).

We now present a sufficient condition for J̃ = ⋃_{x∈X} J_{x} and the complement of the backward images of non-semi-hyperbolic points in fiberwise Julia sets to be of measure zero.

Definition 1.17. (Good points for fibered rational maps) Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y be a fibered rational map over g : X → X. We set J̃_{good}(f) = {z ∈ J̃(f) | lim sup_{n→∞} d(f^{n}(z), UH(f)) > 0}.

Theorem 1.18. (Measure zero) Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y be a fibered rational map over g : X → X. Suppose that all of the following conditions hold:

1. f satisfies condition (C1);

2. for each x ∈ X, the boundary of Ĵ_{x}(f) ∩ UH(f) in Y_{x} does not separate points in Y_{x} (i.e. Y_{x} \ ∂(Ĵ_{x}(f) ∩ UH(f)) is connected);

3. J̃(f) \ ⋃_{n∈N} f^{−n}(UH(f)) ⊂ J̃_{good}(f); and

4. for each z ∈ J̃(f) ∩ UH(f) and each open neighborhood V of z in Y_{π(z)}, the diameter of f_{π(z)}^{n}(V) does not tend to zero as n → ∞.

Then, J̃(f) = ⋃_{x∈X} J_{x} and, for each x ∈ X, the two-dimensional Lebesgue measure of J_{x} \ ⋃_{n∈N} f^{−n}(UH(f)) is equal to zero.

The statement J̃(f) = ⋃_{x∈X} J_{x} in Theorem 1.18 is used to show Theorem 1.25.

We now present a result on rigidity.

Theorem 1.19. (A rigidity) Let (π, Y, X) and (π̃, Ỹ , X̃) be two \overline{ℂ}-bundles. Let f : Y → Y be a fibered rational map over g : X → X and f̃ : Ỹ → Ỹ be a fibered rational map over g̃ : X̃ → X̃. Let u : Y → Ỹ be a homeomorphism that is a bundle conjugacy between f and f̃; i.e., u satisfies the condition that π̃ ∘ u = v ∘ π for some homeomorphism v : X → X and f̃∘ u = u ∘ f. Suppose that f is semi-hyperbolic along fibers and satisfies condition (C1). For each w ∈ X, let u_{w} : Y_{w} → Ỹ_{v(w)} be the restriction of u. Let x ∈ X be a point and assume that u_{x} is K-quasiconformal on F_{x}. Then, for each a ∈ \overline{∪_{n∈ℤ}g^{n}(\{x\})}, it follows that u_{a} : Y_{a} → Ỹ_{v(a)} is K-quasiconformal on the whole Y_{a}, with the same dilatation constant K.

1.2 Results on rational semigroups

In this section, we present several results on rational semigroups. The proofs are given in Section 2.6. Before stating results, we will first establish some notation and definitions regarding the dynamics of rational semigroups.

For a Riemann surface S, let End(S) denote the set of all holomorphic endomorphisms of S. It is a semigroup whose semigroup operation is the functional composition. A rational semigroup is a subsemigroup of End(\overline{ℂ}) without any constant elements. We say that a rational semigroup G is a polynomial semigroup if each element of G is a polynomial. Research on the dynamics of rational semigroups was initiated by A. Hinkkanen and G. J. Martin ([HM1]), who were interested in the role that the dynamics of polynomial semigroups plays in research on various one-complex-dimensional moduli spaces for discrete groups, and by F. Ren’s group([GR]).

In this paper, a rational semigroup G may not be finitely generated, in general.

Definition 1.20. Let G be a rational semigroup. We set

F (G) = \{z ∈ \overline{ℂ} | G is normal in a neighborhood of z\}, J(G) = \overline{ℂ} \ F (G).

F (G) is called the Fatou set of G, and J(G) is called the Julia set of G. The backward orbit G^{−1}(z) of z and the set of exceptional points E(G) are defined by: G^{−1}(z) = ∪_{g∈G}g^{−1}\{z\} and E(G) = \{z ∈ \overline{ℂ} | ♯G^{−1}(z) ≤ 2\}. Furthermore, we set G(z) := ∪_{g∈G}\{g(z)\}. For any subset A of \overline{ℂ}, set G^{−1}(A) = ∪_{g∈G}g^{−1}(A). We denote by ⟨h_{1}, h_{2}, … ⟩ the rational semigroup generated by the family {h_{i}}. For a rational map g, we denote by J(g) the Julia set of the dynamics of g.

We now present a result on uniform perfectness. In the following theorem, we do NOT assume semi-hyperbolicity.

Theorem 1.21. (Uniform perfectness) Let Λ be a compact set in the space \{h : \overline{ℂ} → \overline{ℂ} | h : holomorphic, deg(h) ≥ 2\} endowed with topology induced by uniform convergence on \overline{ℂ}. Let G be a rational semigroup generated by the set Λ. Then there exists a positive constant C such that each subsemigroup H of G satisfies the condition that J(H) is C-uniformly perfect. Furthermore, for any subsemigroup H of G, if a point z_{0} ∈ J(H) satisfies the condition that there exists an element h ∈ H such that h(z_{0}) = z_{0} and h′(z_{0}) = 0, then it follows that z_{0} ∈int J(H).

Example 1.22. Let G = ⟨h_{1}, h_{2}⟩ where h_{1}(z) = 2z + z^{2} and h_{2}(z) = \frac{z^{3}}{z−a}, a ∈ ℂ with a ≠ 0. Then, 0 ∈ J(G) and 0 is a superattracting fixed point of h_{2}. Hence, 0 ∈int J(G), by Theorem 1.21. Since ∞ is a common attracting fixed point of h_{1} and h_{2}, we have ∞ ∈ F (G). Furthermore, let H be a subsemigroup of G such that 0 ∈ J(H) and h_{2} ∈ H. Then, by Theorem 1.21 again, we have 0 ∈int J(H). Moreover, we have ∞ ∈ F (H).

In particular, let H_{0} be a subsemigroup of G that is generated by G \ ⟨h_{1}⟩. Then we have all of the following:

1. 0 ∈ int J(H_{0}).

2. ∞ ∈ F (H_{0}).

3. For any finitely generated subsemigroup H_{1} of H_{0}, we have 0 ∈ F (H_{1}).

For, since 0 ∈int J(G), J(G) = \overline{∪_{g∈G}J(g)} (Corollary 3.1 in [HM1]), and J(h_{1}) is nowhere dense, we obtain that there exists a suquence (g_{n}) in H_{0} such that d(0, J(g_{n})) → 0 as n → ∞. Hence, 0 ∈ J(H_{0}). Since 0 is a superattracting fixed point of h_{2} and h_{2} ∈ H_{0}, by Theorem 1.21 we obtain 0 ∈ int J(H_{0}). Since H_{0} ⊂ G and ∞ ∈ F (G), we obtain ∞ ∈ F (H_{0}). For any finitely generated subsemigroup H_{1} of H_{0}, since 0 is a common attracting fixed point of any element of H_{0}, it follows that 0 ∈ F (H_{1}).

Remark 9. In [HM2], by A. Hinkkanen and G. Martin, it was shown that a finitely generated rational semigroup G such that each g ∈ G is of degree two or greater satisfies the condition that J(G) is uniformly perfect. In [St3], by R. Stankewitz, it was shown that, if Λ (this is allowed to have an element of degree one) is a family of rational maps on \overline{ℂ} such that the Lipschitz constant of each element of Λ with respect to the spherical metric on \overline{ℂ} is uniformly bounded, then the Julia set of semigroup G generated by Λ is uniformly perfect. However, there has been no research done on the uniform perfectness of the Julia sets of subsemigroups of such a semigroup. In [HM2] and [St3], the proofs were based on the density of repelling fixed points in the Julia sets (which was shown by an application of Ahlfors’s five-island theorem), whereas, in this paper, the proof of Theorem 1.21 is based on the combination of the density of repelling fixed points in the Julia set with Proposition 2.2 (potential theory).

We now define the semi-hyperbolicity of rational semigroups.

Definition 1.23. (Semi-hyperbolicity of rational semigroups) Let G be a rational semigroup and N a positive integer. We denote by SH_{N}(G) the set of points z ∈ \overline{ℂ} such that there exists a positive number δ such that, for any g ∈ G and any V ∈ c(B(z, δ), g), it holds that deg(g : V → B(z, δ)) ≤ N. Furthermore, we set UH(G) = \overline{ℂ} \ (∪_{N∈ℕ}SH_{N}(G)). A point z ∈ SH_{N}(G) is called a semi-hyperbolic point of degree N (for G). We say that G is semihyperbolic if J(G) ⊂ ⋃_{N∈ℕ} SH_{N}(G). This is equivalent to J(G) ⊂ SH_{N}(G), for some N ∈ ℕ.

Furthermore, for a rational semigroup G, we set P (G) := \overline{⋃_{g∈G}\{critical values of g\}} (⊂ \overline{ℂ}). This is called the postcritical set of G. We say that G is hyperbolic if P (G) ⊂ F (G). Note that if G is hyperbolic, then G is semi-hyperbolic. Note also that if G = ⟨h_{1}, …, h_{m}⟩, then P (G) =

\overline{⋃_{g∈G} g (⋃_{j=1}^{m}\{ critical values of h_{j}\})}.

Definition 1.24. Let G = ⟨h_{1}, h_{2}, …, h_{m}⟩ be a finitely generated rational semigroup. Let U be a non-empty open set in \overline{ℂ}. We say that G satisfies the open set condition with U with respect to the generator systems {h_{1}, h_{2}, …, h_{m}}, if for each j = 1, …, m, h_{j}^{−1}(U) ⊂ U and {h_{j}^{−1}(U)}_{j=1,... ,m} are mutually disjoint.

We now present a result on porosity.

Theorem 1.25. (Porosity) Let G = ⟨h_{1}, …, h_{m}⟩ be a rational semigroup with an element of degree two or greater. Suppose that all of the following conditions hold:

1. G satisfies the open set condition with an open set U with respect to the generator system {f_{1}, …, f_{m}},

2. ♯(UH(G) ∩ J(G)) < ∞ and

3. UH(G) ∩ J(G) ⊂ U.

Then it follows that J(G) = \overline{U}, or that J(G) is porous (and so the upper box dimension of J(G) is strictly less than 2). Moreover, the fibered rational map f : Σ_{m} × \overline{ℂ} → Σ_{m} × \overline{ℂ} associated with the generator system \{h_{1}, …, h_{m}\} satisfies J̃(f) = ⋃_{x∈Σ_{m}} J_{x}.

We now present results regarding the Hausdorff dimension of Julia sets.

Definition 1.26. Let G be a rational semigroup and δ a non-negative number. We say that a Borel probability measure μ on \overline{ℂ} is δ-subconformal (for G) if, for each g ∈ G and for each Borel-measurable set A, it holds that μ(g(A)) ≤ ∫_{A} ∥g′(z)∥^{δ}dμ, where we denote by ∥ · ∥ the norm of the derivative with respect to the spherical metric. We set s(G) = inf{δ | ∃μ : δ-subconformal measure}.

Let G be a countable rational semigroup. For each x ∈ \overline{ℂ} and each real number s, we set S(s, x) = ∑_{g∈G} ∑_{g(y)=x} ∥g′(y)∥^{−s}, counting multiplicities and S(x) = inf{s | S(s, x) < ∞}. If there is no s such that S(s, x) < ∞, then we set S(x) = ∞. Also, we set s_{0}(G) = inf\{S(x) | x ∈ \overline{ℂ}\}.

An estimate on s_{0}(G) exists when G satisfies the open set condition.

Proposition 1.27. Let G = ⟨h_{1}, …, h_{m}⟩ be a rational semigroup. When m = 1, assume that h_{1} is neither the identity nor an elliptic Möbius transformation. Suppose that G satisfies the open set condition with an open set U with respect to the generator system \{h_{1}, …, h_{m}\}. Suppose also that J(G) ≠ \overline{U}. Then there exists an open set V included in U ∩F (G) such that, for almost every x ∈ V with respect to the two-dimensional Lebesgue measure, S(2, x) < ∞. In particular, s_{0}(G) ≤ 2.

Theorem 1.28. (Hausdorff dimension) Let G = ⟨h_{1}, …, h_{m}⟩ be a rational semigroup. When m = 1, we assume J(G) ≠ \overline{ℂ}. Under the same assumptions as those of Theorem 1.25, it follows that dim_{H}(J(G)) ≤ s(G) ≤ s_{0}(G), where dim_{H} denotes the Hausdorff dimension with respect to the spherical metric in \overline{ℂ}.

Example 1.29. Let h_{1}(z) = z^{2} + 2, h_{2}(z) = z^{2} − 2 and U = {|z| < 2.}. Then, h_{1}^{−1}(U) ∪ h_{2}^{−1}(U) ⊂ U and h_{1}^{−1}(U) ∩ h_{2}^{−1}(U) = ∅. Let h_{3} be a polynomial that is conjugate to h_{4}^{n} by an affine map α, where h_{4}(z) = z^{2} + \frac{1}{4} and n ∈ ℕ is a sufficiently large number. For an appropriate α, it holds that J(h_{3}) ⊂ U \(h_{1}^{−1}(\overline{U})∪h_{2}^{−1}(\overline{U})). For sufficiently large n, h_{3}^{−1}(\overline{U}) ⊂ U \(h_{1}^{−1}(\overline{U})∪h_{2}^{−1}(\overline{U})). Then G = ⟨h_{1}, h_{2}, h_{3}⟩ satisfies the conditions specified as assumptions in Theorem 1.25. In this case, UH(G) ∩ J(G) is one point that is a parabolic fixed point a of h_{3} (For, UH(G) ∩ J(G) is included in P (G) ∩ J(G) = {−2, 2, a}, and, by Lemma 2.14, it follows that \{−2, 2\} ⊂ \overline{ℂ} \ (UH(G) ∩ J(G))) and UH(G) ∩ J(G) ⊂ U. By Theorem 1.25, it can be seen that J(G) is porous and that, in particular, the upper box dimension is strictly less than two. Furthermore, by Theorem 1.28 we obtain that dim_{H}(J(G)) ≤ s(G) ≤ s_{0}(G).

1.3 Remark

We now make some remarks about fibered rational maps and rational semigroups.

Remark 10. 1. Dabija [D] showed that (almost) every holomorphic selfmap f of a ruled surface Y is a fibered rational map over a holomorphic map g : X → X, where X is a Riemann surface.

2. Let p(x) ∈ ℂ[x] be a polynomial of degree two or greater, and q(x, y) ∈ ℂ[x, y] be a polynomial of the form: q(x, y) = y^{n} + a_{1}(x)y^{n−1} + · · · . Let f : ℂ^{2} → ℂ^{2} be a map defined by f((x, y)) = (p(x), q(x, y)). This is called a polynomial skew product in ℂ^{2}. Dynamics of maps of this form were investigated by S.-M. Heinemann in [H1] and [H2], and by M. Jonsson in [J1]. Let X be a compact subset of ℂ such that p(X) ⊂ X. (e.g., the Julia set of p.) Let (π, Y = X × \overline{ℂ}, X) be a trivial \overline{ℂ}-bundle. Then the map f̃: Y → Y defined by f̃((x, y)) = (p(x), q(x, y)) is a fibered rational map over p : X → X.

3. In Corollary 6.7 of [Se3], O. Sester showed that any “non-recurrent quadratic fibered polynomials” with connected fiberwise Julia sets are semi-hyperbolic.

4. Let \{h_{1}, …, h_{m}\} be non-constant rational functions on \overline{ℂ}. Let f : Y → Y be the skew product map associated with the generator system {h_{1}, …, h_{m}}. It can be shown by means of simple arguments that f : Y → Y is semihyperbolic along fibers if and only if G = ⟨h_{1}, …, h_{m}⟩ is semi-hyperbolic.

5. In [S4], with G a finitely generated rational semigroup, a sufficient condition for the semi-hyperbolicity of a point z ∈ J(G) was given, which gives a generalization of R. Mañé’s work([Ma]). Furthermore, in [S4], a necessary and sufficient condition for the semi-hyperbolicity of a finitely generated rational semigroup was given. From this, it can be seen that G = ⟨z^{2} + 2, z^{2} − 2⟩ is semi-hyperbolic, and is not hyperbolic.

2 Tools and Proofs

We now present the proofs of the main results, meanwhile providing further notation and tools.

2.1 Fundamental properties of fibered rational maps

By means of definitions, the following lemma can be easily shown.

Lemma 2.1. Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y be a fibered rational map over g : X → X. Then,

1. For each x ∈ X, f_{x}^{−1}(F_{g(x)}) = F_{x}, f_{x}^{−1}(J_{g(x)}) = J_{x}. Furthermore, f(J̃(f)) ⊂ J̃(f).

2. If g : X → X is an open map, then f^{−1}(J̃(f)) = J̃(f) and f(F̃(f)) ⊂ F̃(f).

3. If g : X → X is a surjective and open map, then f^{−1}(J̃(f)) = J̃(f) = f(J̃(f)) and f^{−1}(F̃(f)) = F̃(f) = f(F̃(f)).

Proof. This proof is the same as that for Lemma 2.4 in [S4].

2.2 Fundamental properties of rational semigroups

For a rational semigroup G, for each f ∈ G, it holds that f(F (G)) ⊂ F (G) and f^{−1}(J(G)) ⊂ J(G). Note that equality f^{−1}(J(G)) = J(G) does not hold in general. If ♯J(G) ≥ 3 , then J(G) is a perfect set, ♯E(G) ≤ 2, J(G) is the smallest closed backward invariant set containing at least three points, and J(G) is the closure of the union of all repelling fixed points of the elements of G, which implies that J(G) = \overline{⋃_{g∈G} J(g)}. If a point z is not in E(G), then, for every x ∈ J(G), x ∈ \overline{G^{−1}(z)}. In particular, if z ∈ J(G) \ E(G) then \overline{G^{−1}(z)} = J(G). For more precise statements, see Lemma 2.3 in [S5], for which the proof is based on [HM1] and [GR]. Furthermore, if G is generated by a precompact subset Λ of End(\overline{ℂ}), then J(G) = \overline{⋃_{f∈Λ} f^{−1}(J(G))} = ⋃_{h∈\overline{Λ}} h^{−1}(J(G)). In particular, if Λ is compact, then J(G) = ⋃_{f∈Λ} f^{−1}(J(G))([S4]). We call this property of the Julia set backward self-similarity.

Remark 11. In the context of backward self-similarity, the existing research on the Julia sets of rational semigroups may be considered as a kind of generalization of the research on self-similar sets constructed by some similarity transformations from ℂ to itself, which can be regarded as the Julia sets of some rational semigroups. It can be easily seen that the Sierpiński gasket is the Julia set of a rational semigroup G = ⟨h_{1}, h_{2}, h_{3}⟩, where h_{i}(z) = 2(z − p_{i}) + p_{i}, i = 1, 2, 3, with p_{1}p_{2}p_{3} a regular triangle.

2.3 Potential theory and measure theory

For the proof of results on uniform perfectness, Johnness, and so on, let us borrow some notation from [J2] and [S4], concerning potential theoretic aspects. By the arguments in [J2] and [S4], for a fibered rational map f : Y → Y over g : X → X with d(x) ≥ 2 for each x ∈ X, one can show a result corresponding to Proposition 2.5 in [S4], using the arguments in §3 in [J2] and from pp. 580-581 in [S4] . In this paper, the following statements, and especially the lower semicontinuity of x ↦ J_{x}(f), are necessary. (Proposition 2.2.3).

Proposition 2.2. Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y be a rational map fibered over g : X → X. Assume that d(x) ≥ 2, for each x ∈ X. Then, for each x ∈ X, there exists a Borel probability measure μ_{x} on Y satisfying all of the following.

1. x ↦ μ_{x} is continuous with respect to the weak topology of probability measures in Y.

2. supp(μ_{x}) = J_{x}, for each x ∈ X.

3. x ↦ J_{x} is lower semicontinuous with respect to the Hausdorff metric in the space of the non-empty compact subsets of Y. That is, if x, x_{n} ∈ X, x_{n} → x as n → ∞ and y ∈ J_{x}, then there exists a sequence (y_{n}) of points in Y with y_{n} ∈ J_{x_{n}}, for each n ∈ ℕ, such that y_{n} → y as n → ∞.

Furthermore , J_{x}(f) is a non-empty perfect set, for each x ∈ X.

Proof. Since d(x) ≥ 2 for each x ∈ X, and x ↦ d(x) is continuous, we can demonstrate these statements in the same way as in §3 in [J2], using the argument from pp. 580-581 in [S4]. The statement 3 follows easily from 1 and 2.

2.4 Distortion lemma and limit functions

For an exposition of the results deduced from semi-hyperbolicity, the following statements are necessary. For the research on the semi-hyperbolicity of the usual dynamics of rational functions, see [CJY] and [Ma]. Notation:

1. Let X be a compact set in \overline{ℂ} and z be a point in \overline{ℂ} \ X. Then we set

Dist(X, z) = max_{y∈X} d(y, z)/ min_{y∈X} d(y, z).

2. For two positive numbers A and B, A ≍ B means K^{−1} ≤ A/B ≤ K, for some constant K independent of A and B.

Lemma 2.3 ([CJY]). (Distortion lemma for proper maps) For any positive integer N and real number r with 0 < r < 1, there exists a constant C = C(N, r) such that, if f : D(0, 1) → D(0, 1) is a proper holomorphic map with deg(f) = N, then H(f(z_{0}), C) ⊂ f(H(z_{0}, r)) ⊂ H(f(z_{0}), r) for any z_{0} ∈ D(0, 1), where H(x, s) denotes the hyperbolic disk in D(0, 1) around x with radius s. Here, the constant C = C(N, r) depends only on N and r.

The following is a generalized distortion lemma for proper maps.

Lemma 2.4 ([S4],[S6]). Let V be a domain in \overline{ℂ}, K, a continuum in \overline{ℂ} with diam_{S}K = a. Assume V ⊂ \overline{ℂ} \ K. Let f : V → D(0, 1) be a proper holomorphic map of degree N. Then there exists a constant r(N, a) depending only on N and a such that, for each r with 0 < r ≤ r(N, a), there exists a constant C = C(N, r) depending only on N and r such that, for each connected component U of f^{−1}(D(0, r)), it follows that diam_{S} U ≤ C, where we denote by diam_{S} the spherical diameter. Furthermore, it follows that C(N, r) → 0 as r → 0.

As the author mentioned in Remark 6 in [S4], the results Lemma 2.13, Theorem 2.14, Lemma 2.15 and Theorem 2.17 in [S4] are generalized to the version of the dynamics of fibered rational maps on \overline{ℂ}-bundles, which we need to show results in this paper. The following lemma is a slightly modified version of Lemma 2.15 in [S4].

Lemma 2.5 ([S4]). Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y be a fibered rational map over g : X → X. Assume that f satisfies condition (C1). Assume z_{0} ∈ SH_{N}(f), for some N ∈ ℕ. Then there exists a positive number δ_{0} such that, for each δ with 0 < δ < δ_{0}, there exists a neighborhood U of x_{0} := π(z_{0}) in X such that, for each n ∈ ℕ, each x ∈ U and each x_{n} ∈ p^{−n}(x), it follows that each element of c(i_{x}i_{x_{0}}^{−1}B̃(z_{0}, δ), f_{x_{n}}^{n}) is simply connected.

The following theorem relates to what happens if there exists a non-constant limit function on a component of a fiber-Fatou set. This is the key to state other results. The proof is the same as that for Lemma 2.13 in [S4].

Theorem 2.6 ([S4]). (Key theorem I) Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y be a fibered rational map over g : X → X. Assume that f satisfies condition (C1). Let z ∈ Y be a point with z ∈ F_{π(z)}. Let (i_{x}) be a local parameterization. Let U be a connected open neighborhood of i_{π(z)}^{−1}(z) in \overline{ℂ}. Suppose that there exists a strictly increasing sequence (n_{j}) of ℕ such that R_{j} := i_{πf^{n_{j}} (z)}^{−1} ∘ f_{π(z)}^{n_{j}} ∘ i_{π(z)} uniformly converges to a non-constant map ϕ on U as j → ∞. Furthermore, suppose that f_{π(z)}^{n_{j}}(z) converges to a point z_{0} ∈ Y. Let

S_{i,j} = f_{g^{n_{i}} π(z)}^{n_{j}−n_{i}} for 1 ≤ i ≤ j. We set

V = {a ∈ Y_{π(z_{0})} | ∃ϵ > 0, lim_{i→∞} sup_{j>i} sup_{ξ∈B̃(a,ϵ)} d(S_{i,j} ∘ φ(ξ), ξ) = 0},

where φ is a map from Y_{π(z_{0})} onto Y_{g^{n_{i}} π(z)} defined by the local parameterization around π(z_{0}). Then V is a non-empty open proper subset of Y_{π(z_{0})}, and it follows that

∂V ⊂ Ĵ_{π(z_{0})}(f) ∩ UH(f).

Corollary 2.7. Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y be a fibered rational map over g : X → X. Assume that f satisfies condition (C1). Assume also that, for each x ∈ X, the boundary of Ĵ_{x}(f) ∩ UH(f) in Y_{x} does not separate points in Y_{x}. Then, for each z ∈ Y with z ∈ F_{π(z)}, it follows that diamf_{π(z)}^{n}(W) → 0 as n → ∞, for each small connected neighborhood W of z in Y_{π(z)}, and that d(f_{π(z)}^{n}(z), UH(f)) → 0 as n → ∞.

Proof. Let z ∈ Y be a point with z ∈ F_{π(z)}. Suppose that the assumptions of Theorem 2.6 are satisfied, and continue with the same notation as that in Theorem 2.6. Let x_{i} = g^{n_{i}}π(z). Let D^{i} be the topological disk in Y_{x_{i}} satisfying condition(C1). We may assume that there exists a non-empty open set A in \overline{ℂ} such that A ⊂ i_{x_{i}}^{−1}(D_{i}) for each i ∈ ℕ. Then, by the definition of V, it must be that i_{π(z_{0})}(A) ⊂ Y_{π(z_{0})} \ V. This contradicts the assumption of the corollary. Hence, it must be that diamf_{π(z)}^{n}(W) → 0 as n → ∞, for each small connected neighborhood W of z in Y_{π(z)}. Then, by Lemma 2.4, it follows that that d(f_{π(z)}^{n}(z), UH(f)) → 0 as n → ∞.

2.5 Proofs of results on fibered rational maps

In this section, we present proofs of the results from Section 1.1. We begin with the following. Proof of Theorem 1.6. First, we prove that there exists a constant C > 0 such that for each x ∈ X, J_{x} is C-uniformly perfect. Since, for each x ∈ X, J_{x} is a non-empty perfect set (Proposition 2.2), it has uncountably many points. Combined with the lower semicontinuity of the map x ↦ J_{x} (statement 3 in Proposition 2.2) and the compactness of X, this suggests that, for any x ∈ X, one can take four points z_{x,1}, z_{x,2}, z_{x,3} and z_{x,4} in J_{x} so that d(z_{x,i}, z_{x,j}) > C_{1}, whenever i ≠ j and x ∈ X, for some constant C_{1} independent of (i, j) and x ∈ X.

Suppose that there exists a sequence of annuli {D_{j}} with D_{j} ⊂ Y_{x_{j}}, x_{j} ∈ X such that D_{j} separates J_{x_{j}}, for each j and mod D_{j} → ∞ as j → ∞. Let D′_{j} and D′′_{j} be the two components of Y_{x_{j}} \ D_{j}. We may assume that

diam D′′_{j} → 0 as j → ∞.  (2)

For, by the existence of {z_{x,i}}_{x,i}, it may be assumed that inf_{j∈ℕ} diam D′_{j} > 0. Then, since mod D_{j} → ∞ as j → ∞, by Lemma 6.1 on p. 34 in [LV] it follows that diam D′′_{j} → 0 as j → ∞.

It may also be assumed that (intD′′_{j}) ∩ J_{x_{j}} ≠ ∅, for each j ∈ ℕ. Hence, there exists a smallest positive integer n_{j} such that diam f^{n_{j}+1}(D′′_{j}) ≥ C_{1}. Then, there exists a constant l_{0} such that l_{0}C_{1} < diam f^{n_{j}}(D′′_{j}), for each j. Since diam f^{n_{j}}(D′′_{j}) < C_{1}, there exist three distinct points in {z_{x′_{j},i}}_{i=1,... ,4} none of which belongs to f^{n_{j}}(D′′_{j}), where x′_{j} = g^{n_{j}}(x_{j}). Since D_{j} ⊂ F_{x_{j}}, it follows that none of these three points belongs to f^{n_{j}}(D_{j}), or to f^{n_{j}}(D_{j} ∪ D′′_{j}). Let φ_{j} : {|z| < 1} → D_{j} ∪ D′′_{j} be a Riemann map such that φ_{j}(0) = y_{j} ∈ D′′_{j} (note that we may assume that int D′_{j} ≠ ∅ for each j ). Then, from the above, it follows that, if we set α_{j} = i_{x′_{j}}^{−1}f^{n_{j}}φ_{j} : \{|z| < 1\} → \overline{ℂ}, then \{α_{j}\}_{j∈ℕ} is normal

in {|z| < 1}. But this causes a contradiction, because diam φ_{j}^{−1}(D′′_{j}) → 0 as j → ∞, which follows from mod φ_{j}^{−1}(D_{j}) = mod D_{j} → ∞ as j → ∞, and l_{0}C_{1} < diam f^{n_{j}}(D′′_{j}), for each j. Hence, we have shown that there exists a constant C > 0 such that for each x ∈ X, J_{x} is C-uniformly perfect.

Next, we prove that there exists a constant C > 0 such that for each x ∈ X, Ĵ_{x} is C-uniformly perfect. Suppose that there exists a sequence of annuli {D_{j}} with D_{j} ⊂ Y_{x_{j}}, x_{j} ∈ X such that D_{j} separates Ĵ_{x_{j}}, for each j, and mod D_{j} → ∞ as j → ∞. Let D′_{j} and D′′_{j} be the two components of Y_{x_{j}} \ D_{j}. As in the previous paragraph, it may be assumed that diam D′′_{j} → 0 as j → ∞.

Fix any j ∈ ℕ. Let y ∈ D′′_{j} ∩ Ĵ_{x_{j}} be a point. There exists a sequence ((x_{j,n}, y_{j,n}))_{n} in X ×Y with y_{j,n} ∈ J_{x_{j,n}}, for each n ∈ ℕ, such that (x_{j,n}, y_{j,n}) → (x, y) as n → ∞. Then we may assume that there exists a number n(j) ∈ ℕ such that i_{x_{j},n(j)}^{−1}(J_{x_{j,n(j)}}) ⊂ i_{x_{j}}^{−1}(D′_{j} ∪ D′′_{j}). Since i_{x_{j},n(j)}^{−1}(J_{x_{j,n(j)}}) ∩ i_{x_{j}}^{−1}(D′′_{j}) ≠ ∅ (take n(j) sufficiently large), by the previous paragraph it must be that i_{x_{j},n(j)}^{−1}(J_{x_{j,n(j)}}) ⊂ i_{x_{j}}^{−1}(D′′_{j}), for large j. However, this contradicts the existence of {z_{x,i}}_{x,i}, because it is also the case that diam D′′_{j} → 0 as j → ∞. Hence, we have proved the first and the second statements in 1. The third statement in 1 follows from the uniform perfectness of J_{x} , the continuity of ω_{x}, and Theorem 7.2 in [Su].

Next, we prove statement 2. Suppose the point z belongs to the boundary of Ĵ_{x} with respect to the topology of Y_{x}, where x = π(z). Under a coordinate exchange, the map f_{x}^{n} around z is conjugate to α(z) = z^{l} for some l ∈ ℕ. Since f_{x}^{n}(Y_{x} \ Ĵ_{x}) ⊂ Y_{x} \ Ĵ_{x}, there exists an annulus A around z in Y_{x} that separates Ĵ_{x} and is isomorphic to a round annulus A′ = {r < |z| < R} in the above coordinate. Then, mod (f_{x}^{ns}(A)) = mod (α^{s}(A′)) → ∞ as s → ∞. In addition, f_{x}^{ns}(A) separates Ĵ_{x}, for each s ∈ ℕ. This contradicts the fact that Ĵ_{x} is uniformly perfect.

Before proving Theorem 1.12, we need the following.

Corollary 2.8. (Corollary of Theorem 1.6) Let (π, Y = X × \overline{ℂ}, X) be a trivial \overline{ℂ}-bundle. Let f : Y → Y be a fibered polynomial map over g : X → X such that d(x) ≥ 2, for each x ∈ X. Let R > 0 be a number such that for each x ∈ X, π\overline{ℂ}(J_{x}(f)) ⊂ \{z | d(z, ∞) > R\}, where π\overline{ℂ} : Y → \overline{ℂ} denotes the projection and d denotes the spherical distance. (Note that such an R exists, since d(x) ≥ 2 for each x ∈ X.) Let ρ_{x}(z)|dz| be the hyperbolic metric on π\overline{ℂ}(A_{x}(f)). Let δ_{x}(z) = inf_{w∈∂(π\overline{ℂ}(A_{x}(f)))} |z − w| for each z ∈ π\overline{ℂ}(A_{x}(f)) ∩ ℂ. Then, there exists a positive constant C depending only on f and R such that for each x ∈ X and each z ∈ π\overline{ℂ}(A_{x}(f)) ∩ \{z | d(z, ∞) > R\}, we have C ≤ ρ_{x}(z)δ_{x}(z) ≤ 1.

Proof. ρ_{x}(z)δ_{x}(z) ≤ 1 follows easily from the Schwarz lemma.

Next, as in the proof of Theorem 1.6, for any x ∈ X, we can take two points z_{x,1} and z_{x,2} in J_{x}(f) so that d(z_{x,1}, z_{x,2}) > c_{0}, whenever x ∈ X, for some positive constant c_{0} independent of x ∈ X. Let ψ_{x}(z) be a Möbius transformation such that ψ_{x}(z_{x,1}) = ∞ and ψ_{x} preserves the spherical metric. Let B_{x} = ψ_{x}(π\overline{ℂ}(A_{x}(f))) (⊂ ℂ). By Theorem 1.6 and Theorem 2.16 in [Su], there exists a positive constant c_{1} such that for each x ∈ X and each z ∈ B_{x}, ρ_{x,1}(z) inf_{w∈∂B_{x}} |z − w| ≥ c_{1}, where ρ_{x,1}(z)|dz| denotes the hyperbolic metric on B_{x}.

Let z ∈ π\overline{ℂ}(A_{x}(f)) ∩ \{y | d(y, z_{x,1}) > c_{0}/2, d(y, ∞) > R\}. Then, we have ρ_{x}(z) = ρ_{x,1}(ψ_{x}(z))|ψ′_{x}(z)|. Since ψ_{x}(z) ∈ {y | d(y, ∞) > c_{0}/2, d(y, ψ_{x}(∞)) > R} and ψ_{x}^{−1}({y | d(y, ψ_{x}(∞)) > R}) ⊂ {y | d(y, ∞) > R}, by the Cauchy formula, we have |ψ′_{x}(z)| = |(ψ_{x}^{−1})′(ψ_{x}(z))|^{−1} ≥ c_{2}, where c_{2} is a positive constant independent of z and x.

Next, let w_{0} ∈ ∂(π\overline{ℂ}(A_{x}(f))) = π\overline{ℂ}(J_{x}(f)) be a point such that δ_{x}(z) = |z − w_{0}|.

Suppose case (1): w_{0} ∈ {y | d(y, z_{x,1}) ≤ c_{0}/4}. Then, |z − w_{0}| ≥ c_{3}, where c_{3} is a positive constant depending only on c_{0}. Further, inf_{w∈∂B_{x}} |ψ_{x}(z) − w| ≤ |ψ_{x}(z) − ψ_{x}(z_{x,2})| ≤ c_{4}, where c_{4} is a positive constant depending only on c_{0}. Hence, δ_{x}(z) ≥ \frac{c_{3}}{c_{4}} inf_{w∈∂B_{x}} |ψ_{x}(z) − w|.

Suppose case (2): w_{0} ∈ {y | d(y, z_{x,1}) > c_{0}/4}. Let γ be the Euclidean segment connecting z and w_{0}. Then, since |w_{0} − z| = δ_{x}(z), we obtain d(z_{x,1}, γ) ≥ c_{5}, where c_{5} is a positive constant depending only on c_{0}, which implies d(∞, ψ_{x}(γ)) ≥ c_{5}. Hence, by the Cauchy formula, we have inf_{w∈∂B_{x}} |ψ_{x}(z) − w| ≤ |ψ_{x}(z) − ψ_{x}(w_{0})| ≤ sup_{w∈γ} |ψ′_{x}(w)| · |z − w_{0}| ≤ c_{6}δ_{x}(z), where c_{6} is a positive constant independent of z and x_{0}.

From these arguments, we find that there exists a positive constant c_{7} depending only on f and R, such that for each z ∈ π\overline{ℂ}(A_{x}(f)) ∩ \{y | d(y, z_{x,1}) > c_{0}/2, d(y, ∞) > R}, we have ρ_{x}(z)δ_{x}(z) ≥ c_{7}. Similarly, we find that there exists a positive constant c_{8} depending only on f and R, such that for each z ∈ π\overline{ℂ}(A_{x}(f)) ∩ \{y | d(y, z_{x,2}) > c_{0}/2, d(y, ∞) > R\}, we have ρ_{x}(z)δ_{x}(z) ≥ c_{8}. Since d(z_{x,1}, z_{x,2}) > c_{0}, we obtain the statement of the corollary.

Proof of Theorem 1.12. This statement can be proved using an approach similar to that in [CJY], using Theorem 2.14 (3) in [S4], Corollary 2.8, Lemma 2.3 and Lemma 2.4.

The procedure is as follows: first, for any (x, y) ∈ X × ℂ, let G_{x}((x, y)) := lim_{n→∞} \frac{1}{d_{n}(x)} log^{+} |π\overline{ℂ}(f_{x}^{n}((x, y)))|, where log^{+}(a) := max\{log a, 0\} for a ∈ ℝ with a > 0. Then, by the arguments in [Se3] (or [Se1]), on A_{x}(f), G_{x} equals the Green’s function with pole (x, ∞), G_{g(x)}(f_{x}(z)) = d(x)G_{x}(z) for any x ∈ X and z ∈ Y_{x}, (x, y) ↦ G_{x}((x, y)) is continuous in X×ℂ, and there exist numbers r > 0 and M > 0 such that for any (x, y) ∈ X ×ℂ with |y| > r, |G_{x}((x, y))−log |y|| ≤ M. Further, for any x ∈ X, we can define a locally conformal map ψ_{x} that is defined on {(x, y) | |y| > R}, where R is a number independent of x ∈ X, such that ψ_{g(x)}f_{x} = (ψ_{x})^{d(x)} and log |ψ_{x}| = G_{x} on {(x, y) | |y| > R}. In particular, G_{x} has no critical points in {(x, y) | |y| > R}.

Next, for each x ∈ X, let ρ_{x}(z)|dz| be the hyperbolic metric on A_{x}(f). Similarly, let δ_{x}(z)|dz| be the quasihyperbolic metric on A_{x}(f) (δ_{x}(z) := inf_{w∈J_{x}} |π\overline{ℂ}(z)− π\overline{ℂ}(w)|). By Theorem 2.14(3) in [S4], we obtain the following claim: Claim 1: There exist constants C > 0 and α > 0 such that for each x ∈ X and each z ∈ A_{x}(f) ∩ (X × ℂ) with δ_{x}(z) ≤ 1, we have δ_{x}(z) ≤ CG_{x}(z)^{α}.

Next, for each x ∈ X and each z ∈ A_{x} ∩ (X × ℂ), we take a Green’s line (a curve of steepest descent for G_{x}) γ_{z} running from z to “J_{x}(f)”, such that f_{x}(γ_{z}) = γ_{f_{x}(z)}. (We choose a direction (left or right) consistently at every critical point encountered.) Then, we show the following: Claim 2: Let 0 < C_{1} < C_{2} < ∞, 0 < C_{3} < C_{4} < ∞ be given numbers. Let τ : [a, b] → A_{x}(f) (0 ≤ a < b < ∞) be a curve that parameterizes part of a Green’s line, such that τ([a, b]) ⊂ {z ∈ A_{x}(f) | C_{1} < δ_{x}(z) < C_{2}} and C_{3} ≤ l_{E}τ < C_{4}, where l_{E} denotes the Euclidean length. Then, there exist constants C_{5} > 0 and 0 < λ_{1} < 1, which depend only on f, C_{1}, C_{2}, C_{3}, and C_{4}, such that |τ(a) − τ(b)| ≥ C_{5} and G_{x}(τ(b)) < λ_{1}G_{x}(τ(a)).

The proof of this claim is immediate, since we assume that either J_{x}(f) is connected for each x ∈ X (then f_{x} and G_{x} have no critical points in A_{x}(f)) or condition 2 in the assumption of Theorem 1.12 is satisfied. Using Claim 2, inf\{d(a, b) | a ∈ π\overline{ℂ}(J_{x}), b ∈ π\overline{ℂ}(A_{x}(f) ∩ C(f)), x ∈ X\} > 0, the Koebe distortion theorem, and Corollary 2.8, we easily obtain the following: Claim 3: Let 0 < C_{6} < C_{7} < ∞ and 0 < C_{8} < ∞ be given numbers. Let τ : [a, b] → A_{x}(f) be a curve that parameterizes part of a Green’s line. Suppose τ([a, b]) ⊂ \{z ∈ A_{x}(f) | |π\overline{ℂ}(z)| ≤ C_{8}\} and C_{6} ≤ l_{ρ_{x}}τ ≤ C_{7} (l_{ρ_{x}} denotes the ρ_{x}-length). Then, there exist constants C_{9} > 0 and 0 < λ_{2} < 1 that depend only on f, C_{6}, C_{7}, and C_{8}, such that C_{9} ≤ d_{ρ_{x}}(τ(a), τ(b)) ≤ C_{7} (d_{ρ_{x}} denotes the distance induced by ρ_{x}(z)|dz|) and G_{x}(τ(b)) < λ_{2}G_{x}(τ(a)).

Next, we will show the following: Claim 4: There exist constants C_{10} > 0, C_{11} > 0, and α > 0 such that for any x ∈ X and any z ∈ A_{x}(f) with G_{x}(z) ≤ C_{10}, we have l_{E}γ_{z} ≤ C_{11}G_{x}(z)^{α}. In particular, any Green’s line γ_{z} lands at a point in J_{x}(f).

To show this claim, take a number C_{10} such that for any x ∈ X, {z ∈ A_{x}(f) | G_{x}(z) ≤ C_{10}} ⊂ {z ∈ A_{x}(f) | δ_{x}(z) ≤ 1}. Next, for any two points w_{1} and w_{2} in a Green’s line γ_{z_{0}} (z_{0} ∈ A_{x}(f)) with G_{x}(w_{2}) ≤ G_{x}(w_{1}), let γ_{z_{0}}(w_{1}, w_{2}) be the arc from w_{1} to w_{2} along γ_{z_{0}}. Let {z_{n}} be a sequence in γ_{z_{0}} such that l_{ρ_{x}}γ_{z_{0}}(z_{n}, z_{n+1}) = c for each n ∈ ℕ, where c is a positive constant that is sufficiently small. Then, by Corollary 2.8, Claim 1, and Claim 3, we obtain l_{E}γ_{z_{0}} = ∑_{n} l_{E}γ_{z_{0}}(z_{n}, z_{n+1}) ≤ Const. ∑_{n} ∫_{γ_{z_{0}} (z_{n},z_{n+1})} δ_{x}(z)ρ_{x}(z)|dz| ≤ Const. ∑_{n} ∫_{γ_{z_{0}} (z_{n},z_{n+1})} δ_{x}(z_{n})ρ_{x}(z)|dz| ≤ Const. ∑_{n} δ_{x}(z_{n}) ≤ Const. ∑_{n} G_{x}(z_{n})^{α} ≤ Const. ∑_{n} λ^{nα}G_{x}(z_{0})^{α} ≤ Const.G_{x}(z_{0})^{α}, where, α > 0 and 0 < λ < 1 are constants and Const. denotes a constant. Hence, we obtain the claim.

Next, we show the following: Claim 5: There exist constants δ′ > 0 and p ∈ ℕ such that for any x ∈ X and any (w, z) ∈ A_{x}(f) × A_{x}(f) with w ∈ γ_{z}, l_{ρ_{x}}γ_{z}(z, w) ≥ p and G_{x}(z) < δ′, we have δ_{x}(w) ≤ \frac{1}{2}δ_{x}(z).

This claim can be shown using the argument on page 11 in [CJY], using Claims 3 and 4, and the distortion lemma for proper holomorphic maps. Next, we show the following: Claim 6: There exist constants δ′ > 0 and C_{12} > 0 such that for any x ∈ X and any (w, z) ∈ A_{x}(f) × A_{x}(f) with w ∈ γ_{z} and G_{x}(z) ≤ δ′, we have δ_{x}(z) ≥ C_{12}|π\overline{ℂ}(z) − π\overline{ℂ}(w)|.

This claim can be shown using Claim 5 and Corollary 2.8. Finally, we show the following: Claim 7: There exists a constant C > 0 such that for any x ∈ X and any (w, z) ∈ A_{x}(f) × A_{x}(f) with w ∈ γ_{z}, we have δ_{x}(z) ≥ C|π\overline{ℂ}(z) − π\overline{ℂ}(w)|.

This claim can easily be shown using Claim 6 and the fact that there exist numbers r > 0 and M > 0 such that for any (x, y) ∈ X × ℂ with |y| > r, |G_{x}((x, y)) − log |y|| ≤ M. Hence, we have shown Theorem 1.12.

Proof of Theorem 1.16. For any y′ ∈ ⋃_{x∈X} J_{x} and r > 0, we set

h(y′, r) = sup{s | ∃y′′ ∈ Y_{π(y′)}, B̃(y′′, s) ⊂ F_{π(y′)} ∩ B̃(y′, r)}

and h(r) = inf{h(y′, r) | y′ ∈ ⋃_{x∈X} J_{x}}. By Theorem 2.14 and Remark 6 in [S4], it follows that J̃(f) = ⋃_{x∈X} J_{x}. Furthermore, by condition (C1), int J_{x} = ∅, for any x ∈ X. Hence, it follows that h(r) > 0 for any r > 0.

Since f is semi-hyperbolic and satisfies condition (C1), by Lemma 2.5 there exists a positive number δ_{1} and a number N ∈ ℕ such that, for any y′ ∈ J̃(f), 0 < δ ≤ δ_{1}, n ∈ ℕ, any V ∈ c(B̃(y′, 2δ), f^{n}), we have that V is simply connected and that deg(f^{n} : V → B̃(y′, 2δ)) ≤ N.

Let y ∈ J̃(f) and r > 0. We set B_{n} = f^{n}(B̃(y, r)) and y_{n} = f^{n}(y), for each n ∈ ℕ. Since y ∈ J_{π(y)}, there exists a smallest positive integer n_{0} such that diam B_{n_{0}+1} > δ_{1}. Then there exists a constant l_{0} such that l_{0}δ_{1} <diam B_{n_{0}}. By Corollary 2.3 in [Y], there exists a constant K that depends only on N and a ball B̃(y_{n_{0}}, r_{0}) ⊂ B_{n_{0}}, with r_{0} ≥diam B_{n_{0}}/K ≥ \frac{l_{0}δ_{1}}{K} such that the element W ∈ c((B̃(y_{n_{0}}, r_{0}), f^{n_{0}}) containing y is a subset of B̃(y, r). There exists a ball B̃(y′, \frac{2}{3}h(r_{0})) included in B̃(y_{n_{0}}, r_{0}) ∩ F_{π(y_{n_{0}} )}.

Let D_{0} be an element of c((B̃(y′, \frac{1}{2}h(r_{0}))), f^{n_{0}}) contained in W. It follows that D_{0} ⊂ F_{π(y)}. Let y′′ ∈ D_{0} ∩(f^{n_{0}})^{−1}{y′} be a point. Then, by Corollaries 1.8 and 1.9 in [S4], it follows that Dist (∂D_{0}, y′′) ≤ M, for some M that depends only on N. Further, by the method used in Corollary 1.9 in [S4] it follows that min_{z∈∂D_{0}} d_{π(y′′)}(y′′, z) ≍ max_{z∈∂E} d_{π(y′′)}(y′′, z), where E ∈ c(B̃(y_{n_{0}}, δ_{1}), f^{n_{0}}) with y′′ ∈ E. Since B̃(y, r) ⊂ E, it follows that diam D_{0} ≍ r. Combining this with Dist (∂D_{0}, y′′) ≤ M leads to the conclusion that there exists a constant 0 < k < 1 that does not depend on y and r such that B(y′′, kr) ⊂ D_{0} ⊂ F_{π(y)}.

To show Theorem 1.18, we present some notation and propositions.

Definition 2.9. (Conical set for fibered rational maps) Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y be a fibered rational map over g : X → X. Let N ∈ ℕ and r > 0. We denote by J̃_{con}(f, N, r) the set of points z ∈ J̃(f) such that, for any ϵ > 0, there exists a positive integer n such that the element U ∈ c(B̃(f^{n}(z), r), f^{n}|_{Y_{π(z)}}) containing z satisfies the following conditions:

1. diam U ≤ ϵ,

2. U is simply connected, and

3. deg(f^{n} : U → B̃(f^{n}(z), r)) ≤ N.

We set J̃_{con}(f, N) = ⋃_{r>0} J̃_{con}(f, N, r) and J̃_{con}(f) = ⋃_{N∈ℕ} J̃_{con}(f, N).

Proposition 2.10. Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y be a fibered rational map over g : X → X. Suppose that Ĵ_{x} has no interior points with respect to the topology in Y_{x}, for each x ∈ X. Then the two-dimensional Lebesgue measure of J̃_{con}(f) ∩ J_{x} is equal to zero, for each x ∈ X.

Proof. Fix N ∈ ℕ. Suppose that there exists a point x ∈ X such that J̃_{con}(f, N)∩ J_{x} has positive measure. Then there exists a Lebesgue density point y ∈ J̃_{con}(f, N) ∩ J_{x}. Let y_{m} = f_{x}^{m}(y) and x_{m} = g^{m}(x), for any m ∈ ℕ. Let δ > 0 be a number such that y ∈ J̃_{con}(f, N, δ). Let U_{m}, U′_{m} be the respective elements of c(B̃(y_{m}, δ/2), f_{x}^{m}), c(B̃(y_{m}, δ), f_{x}^{m}) containing y. Since y ∈ J̃_{con}(f, N, δ), there exists a subsequence (n) in ℕ with n → ∞ such that U′_{n} is simply connected, deg(f_{x}^{n} : U′_{n} → B̃(y_{n}, δ)) ≤ N, for each n, and diam U′_{n} → 0 as n → ∞. By Corollary 1.9 in [S4], for any local parameterization i_{x},

lim_{n→∞} \frac{m(i_{x}^{−1}(U_{n} ∩ J_{x}))}{m(i_{x}^{−1}(U_{n}))} = 1,  (3)

where m denotes the spherical measure of \overline{ℂ}. Using an argument from the proof of Theorem 4.4 in [S4], from (3) we can show that

lim_{n→∞} \frac{m(ı_{x_{n}}^{−1}(B̃(y_{n}, δ/2) ∩ F_{x_{n}}))}{m(i_{x_{n}}^{−1}(B̃(y_{n}, δ/2)))} = 0,  (4)

where i_{x_{n}} denotes a local parameterization. There exists a subsequence (n_{j}) of (n), a point y_{∞} ∈ Y, and a point x_{∞} ∈ X such that y_{n_{j}} → y_{∞} and x_{n_{j}} → x_{∞} as j → ∞. By (4), it follows that B̃(y_{∞}, δ/2) ⊂ Ĵ_{x_{∞}}. On the other hand, by assumption, for any a ∈ X, Ĵ_{a} has no interior point. This is a contradiction.

Proposition 2.11. Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y be a fibered rational map over g : X → X. Suppose that f satisfies condition (C1). Then the following hold.

1.

J̃_{good}(f) ∩ ⋃_{x∈X} J_{x} ⊂ J̃_{con}(f).

2. If it is assumed, furthermore, that, for each x ∈ X, the boundary of Ĵ_{x}(f) ∩ UH(f) in Y_{x} does not separate points in Y_{x}, then

J̃_{good}(f) ⊂ J̃_{con}(f) ∩ ⋃_{x∈X} J_{x}.

Proof. We will first prove the first statement. Let z ∈ ⋃_{x∈X} J_{x} be a point such that lim sup_{n→∞} d(f^{n}(z), UH(f)) > 0. For each m ∈ ℕ, let z_{m} = f^{m}(z) and x_{m} = π(f^{m}(z)). For each m ∈ ℕ and each r > 0, let U_{m}(r), U′_{m}(r) respectively, be the elements of c(B̃(z_{m}, r/2), f_{π(z)}^{m}), c(B̃(z_{m}, r), f_{π(z)}^{m}) containing z. There exists a positive number δ, a positive integer N, and a sequence (n) in ℕ such that deg(f_{π(z)}^{n} : U′_{n}(δ) → B̃(z_{n}, δ)) ≤ N. By Lemma 2.5, taking δ sufficiently small it can be assumed that U′_{n}(δ) is simply connected.

Suppose that diam (U_{n}(δ)) does not tend to zero as n → ∞ in (n). Then, by the distortion lemma for proper maps, there exists a subsequence (n_{j}) of (n) with n_{j} → ∞ and a positive number r such that U_{n_{j}}(δ) ⊃ B̃(z, r), for each j. Hence,

f^{n_{j}}(B̃(z, r)) ⊂ B̃(f_{n_{j}}(z), δ)  (5)

for each j. By condition (C1), if δ is sufficiently small, then (5) contradicts that z ∈ ∪_{x∈X}J_{x}. Hence, it follows that diam U_{n}(δ) → 0 as n → ∞ in (n). Therefore, z ∈ J̃_{con}(f).

The second statement follows from Corollary 2.7 and the first statement.

Corollary 2.12. Let (π, Y, X) be a \overline{ℂ}-bundle. Let f : Y → Y be a fibered rational map over g : X → X. Suppose that J̃(f) = ⋃_{x∈X} J_{x}, and that f satisfies condition (C1). Then, for each x ∈ X, it follows that d(f_{x}^{n}(y), UH(f)) → 0, as n → ∞, for almost every y ∈ J_{x}, with respect to the Lebesgue measure in Y_{x}.

Proof. By condition (C1), Ĵ_{x} = J_{x} has no interior points, for each x ∈ X. By Proposition 2.10 and Proposition 2.11, the following statement results.

Proof of Theorem 1.18. Suppose that there exists a point z ∈ J̃(f) such that z ∈ F_{π(z)}. By Corollary 2.7, for each small connected neighborhood W of z in F_{π(z)}, it follows that diam f^{n}(W) → 0 and d(f^{n}(z), UH(f)) → 0 as n → ∞. However, by conditions three and four in the assumptions of our theorem, this causes a contradiction. Hence, we have shown that J̃(f) = ⋃_{x∈X} J_{x}. By Corollary 2.12, it also follows that the two-dimensional Lebesgue measure of J_{x} \ ⋃_{n∈ℕ} f^{−n}(UH(f)) is equal to zero.

Proof of Theorem 1.19. Let x′ = v(x). We now show the following. Claim: There exists a constant C > 0 such that, for each y ∈ \overline{ℂ},

lim sup_{r→0} sup\{\frac{d_{x′}(u_{x}(i_{x}(y)), u_{x}(i_{x}(y′)))}{d_{x′}(u_{x}(i_{x}(y)), u_{x}(i_{x}(y′′)))} | d(y, y′) = d(y, y′′) = r\} ≤ C, (6)

where d_{x′} is the metric on Ỹ_{x′} induced by the form ω̃_{x′}.

If i_{x}(y) ∈ F_{x}(f), then (6) holds with a constant C = C(K) that depends only on K. Let i_{x}(y) ∈ J_{x}(f) be a point. Let δ_{1} and N ∈ ℕ be the numbers from the proof of Theorem 1.16. Let r > 0 be a number. We set B_{n} = f^{n}(i_{x}(B(y, r))) and y_{n} = f^{n}(i_{x}(y)), for each n ∈ ℕ. Since i_{x}(y) ∈ J_{x}(f), there exists a smallest positive integer n_{0} such that diam B_{n_{0}+1} > δ_{1}. Furthermore, there exists a constant l_{0} such that l_{0}δ_{1} <diam B_{n_{0}}. By Corollary 2.3 in [Y], there exists a constant A that depends only on N, and a ball B̃(y_{n_{0}}, r_{0}) ⊂ B_{n_{0}} with r_{0} ≥diam B_{n_{0}}/A ≥ \frac{l_{0}δ_{1}}{A} such that the component of (f^{n_{0}}i_{x})^{−1}(B̃(y_{n_{0}}, r_{0})) containing y is a subset of B(y, r). Then there exist constants δ′ > 0 and 0 < c < 1 such that

B̃(u(y_{n_{0}}), cδ′) ⊂ u(B̃(y_{n_{0}}, r_{0})) ⊂ u(B_{n_{0}}) ⊂ B̃(u(y_{n_{0}}), δ′).

Since f is semi-hyperbolic and satisfies condition(C1), the same thing holds for f̃. Applying Lemma 2.5, Lemma 2.3, and the Koebe distortion theorem to the map f̃^{n_{0}}, which maps a neighborhood u(i_{x}(B(y, r))) of u(i_{x}(y)) in Ỹ_{x′} to a neighborhood of u(y_{n_{0}}) in Ỹ_{g̃^{n_{0}} (x′)}, it follows that (6), for some constant C′ that does not depend on i_{x}(y) ∈ J_{x}(f). Hence, the claim holds.

By this claim, u_{x} : Y_{x} → Ỹ_{x′} is quasiconformal. Since the two-dimensional Lebesgue measure of J_{x}(f) is zero, which is a consequence of Theorem 1.18, it is K-quasiconformal on Y_{x}. Since f̃_{v(w)} ∘ u_{w} = u_{g(w)} ∘ f_{w} on Y_{w}, and f_{w} and f̃_{v(w)} are holomorphic for each w ∈ X, it follows that u_{a′} : Y_{a′} → Ỹ_{v(a′)} is Kquasiconformal for each a′ ∈ ∪_{n∈ℤ}g^{n}(\{x\}). Hence, for each a ∈ \overline{∪_{n∈ℤ}g^{n}(\{x\})}, the map u_{a} : Y_{a} → Ỹ_{v(a)} is also K-quasiconformal, with the same dilatation constant K.

2.6 Proofs of results on rational semigroups

In this section, we prove the results given in Section 1.2. We now prove Theorem 1.21. Proof of Theorem 1.21. Let f : Λ^{ℕ} × \overline{ℂ} → Λ^{ℕ} × \overline{ℂ} be the fibered rational map over the shift map σ : Λ^{ℕ} → Λ^{ℕ} (σ(h_{1}, h_{2}, h_{3}, …) = (h_{2}, h_{3}, …)) defined as f((h_{1}, h_{2}, …), y) = ((h_{2}, h_{3}, …), h_{1}(y)). Then, by Theorem 1.6, there exist positive constants C_{0} and C_{1} such that each g ∈ G satisfies the conditions that J(g) is C_{0}-uniformly perfect and that diam J(g) > C_{1}.

Let H be any subsemigroup of G. Let A be an annulus that separates J(H). Let V_{1} and V_{2} be two connected components of \overline{ℂ} \ A. Since J(G) = \overline{⋃_{g∈G} J(g)} (Corollary 3.1 in [HM1]) , there exist elements g_{1} and g_{2} in H such that J(g_{1}) ∩ V_{1} ≠ ∅ and J(g_{2}) ∩ V_{2} ≠ ∅.

If J(g_{1}) ∩ V_{2} ≠ ∅ or J(g_{2}) ∩ V_{1} ≠ ∅, then mod A ≤ C_{0}. If J(g_{1}) ∩ V_{2} = ∅ and J(g_{2}) ∩ V_{1} = ∅, then, by Lemma 6.1 on p. 34 in [LV], there exists a constant C = C(C_{1}) that depends only on C_{1} such that mod A ≤ C. The second statement follows from the uniform perfectness of J(H) and Theorem 4.1 in [HM2].

To show Theorem 1.25, we present some notation and lemmas. Notation: Throughout the rest of this section, for a fixed generator system \{h_{1}, …, h_{m}\}, let f : Σ_{m}×\overline{ℂ} → Σ_{m}×\overline{ℂ} be the fibered rational map over the shift map σ : Σ_{m} → Σ_{m}, where Σ_{m} = {1, …, m}^{ℕ}, associated with the generator system \{h_{1}, …, h_{m}\}. We set q_{x}^{(n)}(y) = π\overline{ℂ}(f_{x}^{n}(y)), for any (x, y) ∈ Σ_{m} × \overline{ℂ}.

Lemma 2.13. Let E be a finite subset of \overline{ℂ}. Let ⟨h_{1}, …, h_{m}⟩ be a rational semigroup. Then, for any number M > 0, there exists a positive integer n_{0} such that, for any (n, x, y) ∈ ℕ × Σ_{m} × E with n ≥ n_{0} that satisfies all of the following conditions:

1. q_{x}^{(j)}(y) ∈ E for j = 0, …, n

2. (q_{x}^{(n)})′(y) ≠ 0 and

3. for any i ∈ ℕ and j ∈ ℕ with i + j ≤ n, if q_{σ^{i}(x)}^{(j)}(q_{x}^{(i)}(y)) = q_{x}^{(i)}(y), then

|(q_{σ^{i}(x)}^{(j)})′(q_{x}^{(i)}(y))| > 1,

we have that ∥(q_{x}^{(n)})′(y)∥ > M, where ∥ · ∥ denotes the norm of the derivative with respect to the spherical metric.

Proof. This lemma can be shown by induction on ♯E by means of the same method that was employed in Lemma 1.32 in [S4].

Lemma 2.14. Let G = ⟨h_{1}, …, h_{m}⟩ be a finitely generated rational semigroup. Suppose ♯(UH(G) ∩ J(G)) < ∞ and UH(G) ∩ J(G) ≠ ∅. Then, for each z ∈ UH(G) ∩ J(G), there exists an element g ∈ G, an element h ∈ G, and a point w ∈ UH(G) ∩ J(G) such that h(w) = z, g(w) = w and |g′(w)| ≤ 1.

Proof. Suppose that there exists a point z ∈ UH(G) ∩ J(G) for which there exists no (g, h, w) in the conclusion of our lemma. Then, by Lemma 2.13 and the Koebe distortion theorem, it can easily be seen that, for an arbitrarily small ϵ > 0, there exists a positive number δ and a positive constant N such that, if a point w_{0} ∈ UH(G) ∩ J(G) and an element g_{0} ∈ G satisfy g_{0}(w_{0}) = z, then the diameter of the component V of g_{0}^{−1}(B(z, δ)) containing w_{0} is less than ϵ and deg(g_{0} : V → B(z, δ)) ≤ N. Then, for ϵ sufficiently small, since G is finitely generated and ♯(UH(G) ∩ J(G)) < ∞, we can easily obtain that there exists a positive constant N′ such that, for any element g_{1} ∈ G, and any component W of g_{1}^{−1}(V), it follows that deg(g_{1} : W → V) ≤ N′. This implies that z ∈ SH_{N+N′}(G), and this contradicts that z ∈ UH(G).

In the following Lemma 2.15, Lemma 2.16, and Lemma 2.17, we suppose the assumption of Theorem 1.25. Furthermore, we assume either (1) m ≥ 2, or (2) m = 1 and J(G) ≠ \overline{ℂ}.

Lemma 2.15. If m ≥ 2, then there exists a disk D in F (G) such that

1. \overline{⋃_{g∈G} g(D)} ⊂ F (G) and

2. diam q_{x}^{(n)}(D) → 0 as n → ∞ uniformly on x ∈ Σ_{m}.

In particular, the fibered rational map f satisfies condition (C2).

Proof. Suppose the case m ≥ 2. Let h ∈ G be an element of degree two or greater. Since ∅ ≠ \overline{ℂ} \ \overline{U} ⊂ F (G) and UH(G) ∩ J(G) ⊂ U, there exists an attracting periodic point z_{0} of h in F (G) \ U. Since z_{0} ∈ UH(G) and UH(G) ∩ J(G) ⊂ U again, it follows that there exists a disk D around z_{0} such that \overline{⋃_{g∈G} g(D)} ⊂ F (G). Now the second statement of the lemma results from Lemma 1.30 in [S4].

Lemma 2.16. If UH(G) ∩ J(G) ≠ ∅, then, for each point z ∈ UH(G) ∩ J(G), there exists an element h ∈ G such that z is a parabolic fixed point of h. Furthermore, {β ∈ G | β(z) = z} = {g_{z}^{n} | n ∈ ℕ} for some g_{z} ∈ G such that z is a parabolic fixed point of g_{z}.

Proof. Let z ∈ UH(G) ∩ J(G) be a point. Then z ∈ U. By Lemma 2.14, there exists an element g ∈ G, an element α ∈ G, and a point w ∈ UH(G)∩J(G) such that α(w) = z, g(w) = w, and |g′(w)| ≤ 1. We write α = h_{i_{k}} ⋯ h_{i_{1}} and g = h_{j_{l}} ⋯ h_{j_{1}}. We may assume l ≥ k. Since w ∈ g^{−1}(U) ∩ α^{−1}(U), by the open set condition we obtain (i_{1}, … i_{k}) = (j_{1}, …, j_{k}). Let h := h_{j_{k}} ⋯ h_{j_{1}} · h_{j_{l}} ⋯ h_{j_{k+1}}. Then we easily obtain h(z) = z. Furthermore, |h′(z)| = |g′(w)| ≤ 1.

If deg(h) = 1, then, by Lemma 2.15, it follows that z is a repelling fixed point of h. However, this is a contradiction. If deg(h) ≥ 2, then, since it is assumed that ♯(UH(G)∩J(G)) < ∞, there exists no recurrent critical point c of h in J(h). From [Ma], it follows that z is an attracting or parabolic fixed point of h. Suppose that z is an attracting fixed point of h. Then there exists an open neighborhood V of z in U such that h(V) ⊂ V. Let x ∈ Σ_{m} be a point such that h_{x_{n}} ⋯ h_{x_{1}} = h, for some n, where x = (x_{1}, x_{2}, …). Then, by the open set condition, it follows that h_{x′_{n}} ⋯ h_{x′_{1}}(V) ⊂ \overline{ℂ}\U, for any (x′_{1}, …, x′_{n}) ∈ \{1, …, m\}^{n}\\{(x_{1}, …, x_{n})\}. Hence, G is normal in V, and this is also a contradiction.

Let g_{z} be an element in G_{z} := {β ∈ G | β(z) = z} with minimal word length with respect to the generator system {h_{1}, …, h_{m}}. Let β ∈ G_{z}. Then z ∈ β^{−1}(U) ∩ g_{z}^{−1}(U). By the open set condition, it follows that β = γg_{z} for some γ ∈ G. Then γ ∈ G_{z}. Applying the same argument again to γ, we see that β = g_{z}^{n} for some n ∈ ℕ.

Lemma 2.17. We have that for each (x, y) ∈ π−1 ; \overline{ℂ} (G^{−1}(J(G) \ UH(G))) ∩

J̃(f), lim sup_{n→∞} d(q_{x}^{(n)}(y), UH(G)) > 0.

Proof. Let (x, y) be a point in π−1 ; \overline{ℂ} (G^{−1} (J(G) \ UH(G)))∩J̃(f). Then q_{x}^{(n)}(y) ∈ J(G) \ UH(G), for each n ∈ ℕ.

Assume that lim_{n→∞} d(q_{x}^{(n)}(y), UH(G)) = 0. We now derive a contradiction. Let ϵ > 0 be a small number and let A_{ϵ} be the ϵ-neighborhood of UH(G)∩J(G) in \overline{ℂ}. Then there exists a number n_{0} ∈ ℕ such that q_{x}^{(n)}(y) ∈ A_{ϵ}, for each n ≥ n_{0}.

For each n ≥ n_{0}, let z_{n} ∈ UH(G) ∩ J(G) be the unique point such that d(z_{n}, q_{x}^{(n)}(y)) < ϵ. Since g(UH(G)) ⊂ UH(G), for each g ∈ G, it may be assumed that q_{σ^{n}(x)}^{(1)}(z_{n}) = z_{n+1}, for each n ≥ n_{0}.

Since ♯(UH(G) ∩ J(G)) < ∞, there exists a positive integer n_{1} ≥ n_{0} and l ∈ ℕ such that z_{n_{1}+l} = z_{n_{1}}. Let g_{1} ∈ G be an element such that g_{1}(z_{n_{1}}) = z_{n_{1}}. Let w ∈ {1, …, m}^{l} be the word such that h_{w_{l}} ∘ ⋯ ∘ h_{w_{1}} = g_{1}. Then, by the open set condition, σ^{n_{1}}(x) = w^{∞}. Since we assume that d(q_{x}^{(n)}(y), UH(G)) → 0 as n → ∞, by z_{n_{1}+l} = z_{n_{1}} it follows that g_{1}^{k}(q_{x}^{(n_{1})}(y)) → z_{n_{1}} as k → ∞. Hence, by Lemma 2.16 it must be that z_{n_{1}} is a parabolic fixed point of g_{1}, and q_{x}^{(n_{2})}(y) belongs to W ∩P, where W is a small neighborhood of z_{n_{1}} in U, P is the union of the attracting petals of g_{1} at z_{n_{1}}, and n_{2} is a large positive number with n_{2} ≥ n_{1}. Then there exists an open neighborhood V of y such that q_{x}^{(n_{2})}(V) ⊂ W ∩ P. Taking W sufficiently small and n_{2} sufficiently large, it may be assumed that g_{1}^{s}(q_{x}^{(n_{2})}(V)) ⊂ W ∩P, for any s ∈ ℕ. It follows that q_{x}^{(n)}(V) ⊂ U for each n ∈ ℕ. By the open set condition, for each n ∈ ℕ, it follows that q_{x′}^{(n)}(V) ⊂ \overline{ℂ} \ U, for each x′ ∈ Σ_{m} with (x′_{1}, …, x′_{n}) ∈ {1, …, m}^{n} \ {(x_{1}, …, x_{n})}. Hence, G is normal in V, and this contradicts that y ∈ J(G).

We now present a proof of Theorem 1.25. Proof of Theorem 1.25. By Lemma 2.3 in [S5], we have J(G) ⊂ \overline{U}. Suppose J(G) ≠ \overline{U}. Then, by Proposition 4.3 in [S4], intJ(G) = ∅. For any y′ ∈ J(G) and r > 0, we set

h(y′, r) = sup\{s | ∃y′′ ∈ \overline{ℂ}, B(y′′, s) ⊂ F (G) ∩ B(y′, r) ∩ U\}

and h(r) = inf\{h(y′, r) | y′ ∈ J(G)\}. Then, since int J(G) = ∅ and J(G) ⊂ \overline{U}, it follows that h(r) > 0 for any r > 0.

Let δ_{0} > 0 be a small number. Let B be the δ_{0}-neighborhood of UH(G) ∩ J(G) in \overline{ℂ}. By Lemma 2.5 and Lemma 2.15, there exists a positive number δ_{1} and a number N ∈ ℕ such that, for any y′ ∈ J(G) \ B, 0 < δ ≤ δ_{1} and any V ∈ c(B(y′, 2δ), g), V is simply connected and deg(g : V → B(y′, 2δ)) ≤ N. By Lemma 2.16 and the open set condition, the condition 4 in Theorem 1.18 holds. Hence by Lemma 2.17 and Theorem 1.18, we obtain

J̃(f) = ⋃_{x∈Σ_{m}} J_{x}.  (7)

Let y ∈ J(G) be a point. Since π\overline{ℂ}(J̃(f)) = J(G) (Proposition 3.2 in [S5]), by (7) there exists a point x ∈ Σ_{m} such that y ∈ J_{x}.

Let δ_{2} = min{δ_{0}, δ_{1}}. Let r be a positive number. We set B_{n} = q_{x}^{(n)}(B(y, r)) and y_{n} = q_{x}^{(n)}(y), for each n ∈ ℕ. Since y ∈ J_{x}, there exists a smallest positive integer n_{0} such that diam B_{n_{0}+1} > δ_{2}. Then there exists a constant l_{0} such that l_{0}δ_{2} <diam B_{n_{0}}.

Case 1. y_{n_{0}} ∈ J(G) \ B.

By Corollary 2.3 in [Y], there exists a constant K that depends only on N, and a ball B(y_{n_{0}}, r_{0}) ⊂ B_{n_{0}} with r_{0} ≥diam B_{n_{0}}/K ≥ \frac{l_{0}δ_{2}}{K} such that the element W ∈ c(B(y_{n_{0}}, r_{0}), q_{x}^{(n_{0})}) containing y is a subset of B(y, r). There exists a ball B(y′, \frac{2}{3}h(r_{0})) included in B(y_{n_{0}}, r_{0}) ∩ F (G) ∩ U.

Let D_{0} be the element of c(B(y′, \frac{1}{2}h(r_{0})), q_{x}^{(n_{0})}) contained in W. By the open set condition, we obtain that g^{−1}(U ∩ F (G)) ⊂ U ∩ F (G), for each g ∈ G. Hence, it follows that D_{0} ⊂ F (G) ∩ U. Let y′′ ∈ D_{0} ∩ (q_{x}^{(n_{0})})^{−1}{y′} be a point. Then, by Corollaries 1.8 and 1.9 in [S4], Dist (∂D_{0}, y′′) ≤ M for some M that depends only on N. Further, by the same argument that was used in the proof of Theorem 1.16, diam D_{0} ≍ r. Hence, there exists a constant 0 < k < 1 that does not depend on y and r such that B(y′′, kr) ⊂ D_{0} ⊂ F (G) ∩ B(y, r).

Case 2. y_{n_{0}} ∈ B.

Making use of Lemma 2.16, the fact that UH(G) ∩ J(G) ⊂ U, taking a sufficiently small δ_{0}, and the method used on pp. 286-287 in [Y], one can show that there exists a ball B(y′′, k′r) in B(y, r) ∩ F (G), where k′ is a constant with 0 < k′ < 1 that does not depend on y and r.

We now prove Proposition 1.27. Proof of Proposition 1.27. By the open set condition and Lemma 2.3 (f) in [S5], J(G) ⊂ \overline{U}. We now show the following.

Claim 1: There exists an open set V′ included in U ∩ F (G) such that h^{−1}(V′) ∩ V′ = ∅, for each h ∈ G.

Before supporting this claim, we would like the reader to note that the following claim can easily be shown.

Claim 2: If there exists a point z ∈ U ∩ F (G) such that z ∈ \overline{ℂ} \ \overline{G(z)}, then Claim 1 holds with a small open neighborhood V′ of z.

As a proof of Claim 1, by the open set condition, we obtain that

⋃_{j=1}^{m} h_{j}^{−1}(U ∩ F (G)) ⊂ U ∩ F (G).  (8)

Suppose the equality does not hold in (8). Then there exists a point z ∈ U∩F (G) such that h_{j}(z) ∈ \overline{ℂ}\U for each j = 1, …, m. Hence, by the open set condition, z ∈ \overline{ℂ} \ \overline{G(z)}. By Claim 2, Claim 1 holds.

Hence, it may be assumed that

⋃_{j=1}^{m} h_{j}^{−1}(U ∩ F (G)) = U ∩ F (G).  (9)

Let α : U ∩ F (G) → U ∩ F (G) be the map defined as: α(z) = h_{j}(z) if z ∈ h_{j}^{−1}(U ∩ F (G)). This is well defined by (9) and the open set condition.

Let z ∈ U ∩ F (G) be a point. If z ∈ \overline{ℂ} \ \overline{G(z)}, then, by Claim 2, Claim 1 follows. Hence, it may be assumed that z ∈ \overline{G(z)}; i.e.,

z ∈ \overline{⋃_{n=0}^{∞}\{α^{n}(z)\}}.  (10)

Let W be the connected component of U ∩ F (G) containing z. By (10), there exists a smallest positive integer n with α^{n}(W) ⊂ W. By (10) and the open set condition, one of the following two cases holds.

Case 1: W is included in an attracting or a parabolic basin of an element g ∈ G, z is a fixed point in the basin, and g|_{W} = α^{n}|_{W} .

Case 2: W is included in a Siegel disk or a Herman ring of an element g ∈ G of degree 2 or greater, and g|_{W} = α^{n}|_{W} .

If Case 1 holds, then there exists an open set V′ included in W with α^{−l}(V′)∩ V′ = ∅, for each l ∈ ℕ; i.e., h^{−1}(V′) ∩ V′ = ∅, for each h ∈ G.

If Case 2 holds, then, for V′ in a connected component A of α^{−n}(W) with A∩W = ∅, it follows that α^{−l}(V′)∩V′ = ∅, for each l ∈ ℕ; i.e., h^{−1}(V′)∩V′ = ∅, for each h ∈ G.

Hence, we have proved Claim 1. Let V′ be an open set included in U ∩F (G) such that h^{−1}(V′) ∩ V′ = ∅ for each h ∈ G. Then, by the open set condition, g^{−1}(V′) ∩ h^{−1}(V′) = ∅, if g, h ∈ G and g ≠ h. Furthermore, the postcritical

set P (G) of G, which equals \overline{⋃_{g∈G} g (⋃_{j=1}^{m}\{critical values of h_{j}\})}, does not accumulate in V′. Let V be an open disk included in V′ \ P (G). Then, it follows that ∫_{V} ∑_{h∈G} ∑_{α} ∥α′(z)∥^{2} dm(z) < ∞, where α runs over all the well-defined

inverse branches of h on V. Hence, for almost every x ∈ V with respect to the Lebesgue measure, it follows that S(2, x) < ∞.

We now prove Theorem 1.28. We first present several lemmas.

Lemma 2.18. Let G be a rational semigroup. Assume that ∞ ∈ F (G) and that, for each x ∈ E(G), there exists an element g ∈ G such that g(x) = x and |g′(x)| < 1. Let A be a subset of J(G). Suppose that there exist positive constants a_{1} , a_{2} and c with 0 < c < 1 such that, for each x ∈ A, there exist two sequences (r_{n}) and (R_{n}) of positive real numbers and a sequence (g_{n}) of the elements of G satisfying all of the following conditions:

1. r_{n} → 0 and, for each n, 0 < \frac{r_{n}}{R_{n}} < c and g_{n}(x) ∈ J(G).

2. for each n, g_{n}(D(x, R_{n})) ⊂ D(g_{n}(x), a_{1}).

3. for each n g_{n}(D(x, r_{n})) ⊃ D(g_{n}(x), a_{2}).

Then, dim_{H}(A) ≤ s(G).

Proof. We may assume that ♯(J(G)) ≥ 3. Let δ ≥ s(G) be a number and μ a δ-subconformal measure. Using the method employed in the proof of Lemma 5.5 in [S4], one can show that there exists a constant c′ > 0 that does not depend on n ∈ ℕ and x ∈ A such that μ(D(x, r_{n})) ≥ c′r_{n}^{δ}. From this and Theorem 7.2 in [Pe], it follows that dim_{H}A ≤ δ.

Definition 2.19. (Conical set for rational semigroups) Let G be a rational semigroup. Let N ∈ ℕ and r > 0. We denote by J_{con}(G, N, r) the set of points z ∈ J(G) such that, for any ϵ > 0, there exists an element g ∈ G such that g(z) ∈ J(G) and the element U ∈ c(B(g(z), r), g) containing z satisfies the following conditions:

1. diam U ≤ ϵ,

2. U is simply connected, and

3. deg(g : U → B(g(z), r)) ≤ N.

We set J_{con}(G, N) = ∪_{r>0}J_{con}(G, N, r) and J_{con}(G) = ∪_{N∈ℕ}J_{con}(G, N).

Proposition 2.20. Let G be a rational semigroup. Assume that F (G) ≠ ∅ and that, for each x ∈ E(G), there exists an element g ∈ G such that g(x) = x and |g′(x)| < 1. Then, dim_{H}(J_{con}(G)) ≤ s(G).

Proof. We now have only to show the following:

Claim: For fixed N ∈ ℕ and r > 0, dim_{H}(J_{con}(G, N, r)) ≤ s(G).

We may assume that ∞ ∈ F (G). Let x ∈ J_{con}(G, N, r) be a point. Then there exists a sequence (g_{n}) in G such that, for each n ∈ ℕ, we have that g_{n}(x) ∈ J(G), deg(g_{n} : V_{n}(r) → D(g_{n}(x), r)) ≤ N, V_{n}(r) is simply connected and diam V_{n}(r) → 0 as n → ∞, where V_{n}(r) is the element of c(D(g_{n}(x), r), g_{n}) containing x. Let φ_{n} : D(0, 1) → V_{n}(r) be the Riemann map such that φ_{n}(0) = x. By the Koebe distortion theorem, for each n, V_{n}(r) ⊃ D(x, \frac{1}{4}|φ′_{n}(0)|). By Lemma 2.3 and the Koebe distortion theorem, there exists an ϵ > 0 such that, for each n ∈ ℕ, V_{n}(ϵr) ⊂ D(x, \frac{1}{8}|φ′_{n}(0)|). Since diam V_{n}(r) → 0 as n → ∞, it follows that |φ′_{n}(0)| → 0 as n → ∞. The application of Lemma 2.18 proves the claim.

Definition 2.21. (Good points for finitely generated rational semigroups) Let G = ⟨h_{1}, …, h_{m}⟩ be a rational semigroup. Let f : Σ_{m} × \overline{ℂ} → Σ_{m} × \overline{ℂ} be the fibered rational map associated with the generator system \{h_{1}, …, h_{m}\}. Then we set J_{good}(G) = π\overline{ℂ}(J̃_{good}(f)). Note that this definition does not depend on the choice of any generator system of G that consists of finitely many elements.

We now prove the following theorem.

Theorem 2.22. Let G = ⟨h_{1}, …, h_{m}⟩ be a finitely generated rational semigroup with F (G) ≠ ∅. Let f : Y → Y be the fibered rational map associated with the generator system \{h_{1}, …, h_{m}\}, where Y = Σ_{m} × \overline{ℂ}. Suppose that f satisfies condition (C1) and that, for each x ∈ Σ_{m}, the boundary of Ĵ_{x}(f) ∩ UH(f) in Y_{x} does not separate points in Y_{x}. Then, J_{good}(G) ⊂ J_{con}(G) and dim_{H}(J_{good}(G)) ≤ s(G) ≤ s_{0}(G).

Proof. We may assume that ♯(J(G)) ≥ 3. We now show the following:

Claim: If E(G) ≠ ∅, then, for each x ∈ E(G), there exists an element g ∈ G such that g(x) = x and |g′(x)| < 1.

If there exists an element h ∈ G with deg(h) ≥ 2, then this claim is trivial. Suppose that each element of G is of degree one. By Lemma 2.3 (d) in [S5], ♯(E(G)) ≤ 2. Since f satisfies condition (C1), for each i, h_{i} is loxodromic. Since h_{i}(E(G)) = E(G), for each i, it must be that each x ∈ E(G) is fixed by h_{i}, for each i. Let x ∈ E(G) be a point. Suppose |h′_{i}(x)| > 1, for each i. Then, J(G) = {x}, and this is a contradiction, since we assume that ♯(J(G)) ≥ 3. Hence, |h′_{i}(x)| < 1 for some i. Hence, the claim holds.

The statement of this theorem follows from the claim, the second statement in Proposition 2.11, Proposition 2.20 and Theorem 4.2 in [S2].

We now prove Theorem 1.28. Proof of Theorem 1.28. This follows from Lemma 2.15, π\overline{ℂ}(J̃(f)) = J(G) (Proposition 3.2 in [S5]), Lemma 2.17 and Theorem 2.22, when m ≥ 2.

We now suppose m = 1 and J(G) ≠ \overline{ℂ}. Then, by [Ma], J_{good}(G) ⊂ J_{con}(G). By Lemma 2.17 and Proposition 2.20, it follows that dim_{H}(J(G)) ≤ s(G). Moreover, by Theorem 4.2 in [S2], we have s(G) ≤ s_{0}(G).

Hence, we have proved Theorem 1.28.

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