Dimensions of Julia sets of expanding 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

Dedicated to the memory of Professor Nobuyuki Suita

September 23, 2005

Abstract

We estimate the upper box and Hausdorff dimensions of the Julia set of an expanding semigroup generated by finitely many rational functions, using the thermodynamic formalism in ergodic theory. Furthermore, we show Bowen’s formula, and the existence and uniqueness of a conformal measure, for a finitely generated expanding semigroup satisfying the open set condition.

1 Introduction

For a Riemann surface S, let End(S) denote the set of all holomorphic endomorphisms of S. This 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 of the dynamics of polynomial semigroups while studying various one-complex-dimensional moduli spaces for discrete groups, and F. Ren’s group([ZR], [GR]). For references on research into rational semigroups, see [HM1], [HM2], [HM3], [ZR], [GR], [SSS], [Bo], [St1], [St2], [St3], [S1], [S2], [S3], [S4], [S5], [S6], and [S7]. The research on the dynamics of rational semigroups can be considered a generalization of studies of both the iteration of rational functions and self-similar sets constructed using iterated function systems of some similarity transformations in ℝ^{2} in fractal geometry. In both fields, the estimate of the upper(resp.lower) box dimension, which is denoted by \overline{dim}_{B} (resp.\underline{dim}_{B}), and the Hausdorff dimension, which is denoted by dim_{H}, of the invariant sets (Julia sets or attractors) has been of great interest and has been investigated for a long time. In this paper, we consider the following: For a rational semigroup G, We set

∗Published in Kodai Mathematical Journal, Vol. 28, No.2 (2005), pp. 390-422. The title was changed. The first version (of which title was ”The dimensions of Julia sets of expanding rational semigroups”) was written on May 27,2004. Keywords: expanding rational semigroup, Hausdorff dimension of a Julia set, thermodynamic formalism.

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

F (G) is called the Fatou set for G and J(G) is called the Julia set for G. We use ⟨f_{1}, f_{2}, ⋯ ⟩ to denote the rational semigroup generated by the family {f_{i}}. For a finitely generated rational semigroup G = ⟨f_{1}, ⋯, f_{m}⟩, we set Σ_{m} = {1, ⋯, m}^{ℕ} (this is a compact metric space) and we use σ : Σ_{m} → Σ_{m} to denote the shift map, which is (w_{1}, w_{2}, ⋯) ↦ (w_{2}, w_{3}, ⋯) for w = (w_{1}, w_{2}, w_{3}, ⋯) ∈ Σ_{m}. We define the map f̃: Σ_{m} × \overline{ℂ} → Σ_{m} × \overline{ℂ} using

f̃((w, x)) = (σw, f_{w_{1}}(x)).

We call map f̃ the skew product map associated with the generator system {f_{1}, ⋯, f_{m}}. For each w ∈ Σ_{m}, we use F_{w} to denote the set of all the points x ∈ \overline{ℂ} that satisfy the fact that there exists an open neighborhood U of x such that the family \{f_{w_{n}} ∘ ⋯ ∘ f_{w_{1}}\}_{n} is normal in U. We set J_{w} = \overline{ℂ} \ F_{w} and J̃_{w} = {w} × J_{w}. Moreover, we set

J̃(f̃) = \overline{⋃_{w∈Σ_{m}} J̃_{w}}, F̃(f̃) = (Σ_{m} × \overline{ℂ}) \ J̃(f̃),

where the closure is taken in the product space Σ_{m} ×\overline{ℂ} (this is a compact metric space). We call F̃(f̃) the Fatou set for f̃ and J̃(f̃) the Julia set for f̃. For each (w, x) ∈ Σ_{m} × \overline{ℂ} and n ∈ ℕ we set

(f̃^{n})′((w, x)) = (f_{w_{n}} ⋯ f_{w_{1}})′(x).

Furthermore, we denote the first (resp. second) projection by π : Σ_{m} ×\overline{ℂ} → Σ_{m} (resp. π\overline{ℂ} : Σ_{m} × \overline{ℂ} → \overline{ℂ}). We say that a finitely generated rational semigroup G = ⟨f_{1}, ⋯, f_{m}⟩ is an expanding rational semigroup if J(G) ≠ ∅ and the skew product map f̃: Σ_{m} × \overline{ℂ} → Σ_{m} × \overline{ℂ} associated with the generator system {f_{1}, ⋯, f_{m}} is expanding along fibers, i.e., there exists a positive constant C and a constant λ > 1 such that for each n ∈ ℕ,

inf_{z∈J̃(f̃)} ∥(f̃^{n})′(z)∥ ≥ Cλ^{n},

where we use ∥ · ∥ to denote the norm of the derivative with respect to the spherical metric.

For a general rational semigroup G and a non-negative number t, we say that a Borel probability measure τ on \overline{ℂ} is t-subconformal (for G) if for each g ∈ G and for each Borel measurable set A in \overline{ℂ}, τ(g(A)) ≤ ∫_{A} ∥g′∥^{t}dτ. Moreover, we set s(G) = inf{t | ∃τ : t-subconformal measure}.

Furthermore, we say that a Borel probability measure τ on J(G) is t-conformal (for G) if for any Borel set A and g ∈ G, if A, g(A) ⊂ J(G) and g : A → g(A) is injective, then τ(g(A)) = ∫_{A} ∥g′∥^{t} dτ.

For any s ≥ 0 and x ∈ \overline{ℂ}, we set S(s, x) = ∑_{g∈G} ∑_{g(y)=x} ∥g′(y)∥^{−s}. Moreover, we set S(x) = inf{s ≥ 0 | S(s, x) < ∞} (If no s exists with S(s, x) < ∞, then we set S(x) = ∞). We set s_{0}(G) = inf\{S(x) | x ∈ \overline{ℂ}\}. Note that if G has only countably many elements, then s(G) ≤ s_{0}(G) (Theorem 4.2 in [S2].)

Then, under the above notations, we show the following:

Theorem 1.1. (Main Theorem A) Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated expanding rational semigroup. Let f̃ : Σ_{m} × \overline{ℂ} → Σ_{m} × \overline{ℂ} be the skew product map associated with {f_{1}, ⋯, f_{m}}. Then, there exists a unique zero δ of the function: P (t) := P (f̃|_{J̃(f̃)}, tφ̃), where φ̃ is the function on J̃(f̃) defined by: φ̃((w, x)) = − log(∥(f_{w_{1}})′(x)∥) for (w, x) = ((w_{1}, w_{2}, ⋯), x) ∈ J̃(f̃) and P ( , ) denotes the pressure. Furthermore, δ satisfies the fact that there exists a unique probability measure ν̃ on J̃(f̃) such that M_{δ}^{∗}ν̃ = ν̃, where M_{δ} is an operator on C(J̃(f̃)) (the space of continuous functions on J̃(f)) defined by

M_{δ}ψ((w, x)) = ∑_{f̃((w′,y))=(w,x)} \frac{ψ((w′, y))}{∥(f_{w′_{1}})′(y)∥^{δ}} ,

where w′ = (w′_{1}, w′_{2}, ⋯) ∈ Σ_{m}. Moreover, δ satisfies

\overline{dim}_{B}(J(G)) ≤ s(G) ≤ s_{0}(G) ≤ δ = \frac{h_{αν̃}(f̃)}{− ∫_{J̃(f̃)} φ̃αdν̃} ≤ \frac{log(∑_{j=1}^{m} deg(f_{j}))}{− ∫_{J̃(f̃)} φ̃αdν̃},

where α = lim_{l→∞} M_{δ}^{l}(1) and we denote the metric entropy of (f̃, αν̃) by h_{αν̃}(f̃). The support for ν := (π\overline{ℂ})_{∗}ν̃ equals J(G).

Furthermore, let A(G) = \overline{∪_{g∈G}g(\{x ∈ \overline{ℂ} | ∃h ∈ G, h(x) = x, |h′(x)| < 1\})} and P (G) = \overline{⋃_{g∈G}\{all critical values of g\}}. Then, A(G)∪P (G) ⊂ F (G) and for each x ∈ \overline{ℂ} \ (A(G) ∪ P (G)), we have δ is equal to:

inf{t ≥ 0 | ∑_{n∈ℕ} ∑_{(w_{1},··· ,w_{n})∈{1,··· ,m}^{n}} ∑_{(f_{w_{1}} ···f_{wn})(y)=x} ∥(f_{w_{1}} ⋯ f_{w_{n}})′(y)∥^{−t} < ∞}.

Theorem 1.2. (Main Theorem B) Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated expanding rational semigroup. Suppose that there exists a non-empty open set U in \overline{ℂ} such that f_{j}^{−1}(U) ⊂ U for each j = 1, ⋯, m and \{f_{j}^{−1}(U)\}_{j=1}^{m} are mutually disjoint. Then, we have the following:

1. dim_{H}(J(G)) = \overline{dim}_{B}(J(G)) = s(G) = s_{0}(G) = δ, where δ denotes the number in Theorem 1.1.

2. ν := (π\overline{ℂ})_{∗}ν̃ is the unique δ-conformal measure, where ν̃ is the measure in Theorem 1.1. Furthermore, ν satisfies the fact that there exists a positive constant C such that for any x ∈ J(G) and any positive number r with r < diam \overline{ℂ}, we have

C^{−1} ≤ \frac{ν(B(x, r))}{r^{δ}} ≤ C.

3. ν satisfies ν(f_{i}^{−1}(J(G)) ∩ f_{j}^{−1}(J(G))) = 0, for each i, j ∈ {1, ⋯, m} with i ≠ j. Furthermore, for each (i, j) with i ≠ j, we have f_{i}^{−1}(J(G)) ∩ f_{j}^{−1}(J(G)) is nowhere dense in f_{j}^{−1}(J(G)).

4. 0 < H^{δ}(J(G)) < ∞, where H^{δ} denotes the δ-dimensional Hausdorff measure with respect to the spherical metric. Furthermore, we have ν = \frac{H^{δ}|_{J(G)}}{H^{δ}(J(G))}.

5. If there exists a t-conformal measure τ, then t = δ and τ = ν.

6. For any x ∈ \overline{ℂ} \ (A(G) ∪ P (G)), we have

dim_{H}(J(G)) = δ = inf{t ≥ 0 | ∑_{g∈G} ∑_{g(y)=x} ∥g′(y)∥^{−t} < ∞}.

Remark 1. In [S6], it is shown that if G = ⟨f_{1} ⋯, f_{m}⟩ is expanding and there exists a non-empty open set U such that f_{j}^{−1}(U) ⊂ U for each j = 1, ⋯, m, \{f_{j}(U)\}_{j} are mutually disjoint and \overline{U} ≠ J(G), then J(G) is porous and \overline{dim}_{B}(J(G)) < 2.

Remark 2. In addition to the assumption of Main Theorem B, if J(G) ⊂ ℂ, then we can also show a similar result for the Euclidean metric.

For the precise notation, see the following sections. The proof of Main Theorem A is given in section 3 and the proof of Main Theorem B is given in section 5. The existence of a subconformal or conformal measure is deduced by applying some of the results in [W1] and the thermodynamic formalism in ergodic theory to the skew product map associated with the generator system. Since generator maps are not injective in general and we do not assume the “cone condition” (the existence of uniform cones) for the boundary of the open set, much effort is needed to estimate ν(B(x, r)) in Main Theorem B. Indeed, we cut the closure of the open set into small pieces {K_{j}}, and for a fixed s ∈ ℕ, let K be the set of all (γ, k_{j}) that satisfies that γ is a well defined inverse branch of (f_{w_{1}} ∘ ⋯ f_{w_{u}})^{−1} defined on K_{j} for some (w_{1}, ⋯, w_{u}) ∈ {1, ⋯, m}^{u} with u ≤ s. Then we introduce an equivalence class “ ∼ ” in a subset Γ of K, and an order “ ≤ ” in Γ/ ∼ . We obtain an upper estimate of the cardinality of the set of all minimal elements of (Γ/∼, ≤) by a constant independent of r and x, which gives us the key to estimate ν(B(x, r)).

Note that in [MU1], it was discussed the case in which there are infinitely many injective generator maps and the boundary of the open set satisfies the cone condition.

The uniqueness of a conformal measure τ is deduced from some results in [W1] and an estimate of τ(B(x, r)). Note that our definition of conformal measure differs from that of [MU1] and [MU2]. In this paper, we do not require the separating condition for the definition of conformal measure.

2 Preliminaries

In this section, we give the notation and definitions for rational semigroups and the associated skew products that we need to give our main result.

2.1 Rational semigroups

We use the definition in [S5].

Definition 2.1. 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 for G and J(G) is called the Julia set for 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\}. For any subset A of \overline{ℂ}, we set G^{−1}(A) = ∪_{g∈G}g^{−1}(A). We use ⟨f_{1}, f_{2}, ⋯ ⟩ to denote the rational semigroup generated by the family {f_{i}}. For a rational map g, we use J(g) to denote the Julia set of dynamics of g.

For a rational semigroup G, for each f ∈ G, we have f(F (G)) ⊂ F (G) and f^{−1}(J(G)) ⊂ J(G). Note that we do not have this equality 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 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). Further, for a finitely generated rational semigroup G = ⟨f_{1}, ⋯, f_{m}⟩, if we use G_{n} to denote the subsemigroup of G that is generated by n-products of generators {f_{j}}, then J(G_{n}) = J(G). For more precise statements, see Lemma 2.3 in [S5], for which the proofs are 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 we have J(G) = ⋃_{f∈Λ} f^{−1}(J(G))([S3]). We call this property of a Julia set the backward self-similarity.

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

2.2 Associated skew products

We use the notation in [S5]. Let m be a positive integer. We use Σ_{m} to denote the one-sided wordspace that is Σ_{m} = {1, ⋯, m}^{ℕ} and use σ : Σ_{m} → Σ_{m} to denote the shift map, which is (w_{1}, ⋯) ↦ (w_{2}, ⋯) for w = (w_{1}, w_{2}, w_{3}, ⋯) ∈ Σ_{m}. For any w, w′ ∈ Σ_{m}, we set d(w, w′) := ∑_{n=1}^{∞}(1/2^{n}) · c(w_{k}, w′_{k}), where c(w_{k}, w′_{k}) = 0 if w_{k} = w′_{k} and c(w_{k}, w′_{k}) = 1 if w_{k} ≠ w′_{k}. Then, (Σ_{m}, d) is a compact metric space. Furthermore, the dynamics of σ : Σ_{m} → Σ_{m} is expanding with respect to this metric d. That is, each inverse branch σ_{j}^{−1} of σ^{−1} on Σ_{m}, which is defined by σ_{j}^{−1}((w_{1}, w_{2}, ⋯)) = (j, w_{1}, w_{2}, ⋯) for j = 1, ⋯, m, satisfies d(σ_{j}^{−1}(w), σ_{j}^{−1}(w′)) ≤ (1/2) · d(w, w′).

Let G = ⟨f_{1}, f_{2}, ⋯, f_{m}⟩ be a finitely generated rational semigroup. We define the map f̃: Σ_{m} × \overline{ℂ} → Σ_{m} × \overline{ℂ} using

f̃((w, x)) = (σw, f_{w_{1}}(x)).

We call map f̃ the skew product map associated with the generator system {f_{1}, ⋯, f_{m}}. f̃ is finite-to-one and an open map. We hold that point (w, x) ∈ Σ_{m} × \overline{ℂ} satisfies f′_{w_{1}}(x) ≠ 0 if and only if f̃ is a homeomorphism in a small neighborhood of (w, x). Hence, the map f̃ has infinitely many critical points in general.

Definition 2.2. For each w ∈ Σ_{m}, we use F_{w} to denote the set of all the points x ∈ \overline{ℂ} that satisfy the fact that there exists an open neighborhood U of x such that the family \{f_{w_{n}} ∘ ⋯ ∘ f_{w_{1}}\}_{n} is normal in U. We set J_{w} = \overline{ℂ} \ F_{w} and J̃_{w} = {w} × J_{w}. Moreover, we set

J̃(f̃) = \overline{⋃_{w∈Σ_{m}} J̃_{w}}, F̃(f̃) = (Σ_{m} × \overline{ℂ}) \ J̃(f̃),

where the closure is taken in the product space Σ_{m} × \overline{ℂ}. We often write F̃(f̃) as F̃ and J̃(f̃) as J̃. We call F̃(f̃) the Fatou set for f̃ and J̃(f̃) the Julia set for f̃. Here, we remark that ⋃_{w∈Σ_{m}} J̃_{w} may not be compact in general. That is why we consider the closure of that set in Σ_{m} × \overline{ℂ} (this is a compact space) concerning the definition of the Julia set for f̃.

For each (w, x) ∈ Σ_{m} × \overline{ℂ} and n ∈ ℕ we set

(f̃^{n})′((w, x)) = (f_{w_{n}} ⋯ f_{w_{1}})′(x).

Furthermore, we denote the first (resp. second) projection by π : Σ_{m} ×\overline{ℂ} → Σ_{m} (resp. π\overline{ℂ} : Σ_{m} × \overline{ℂ} → \overline{ℂ}). Note that we have f̃(F̃(f̃)) = f̃^{−1}(F̃(f̃)) = F̃(f̃), f̃(J̃(f̃)) = f̃^{−1}(J̃(f̃)) = J̃(f̃) and π\overline{ℂ}(J̃(f̃)) = J(G). (For the fundamental properties of these sets, see Proposition 3.2 in [S5]. In addition, see [S3].)

Definition 2.3. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated rational semigroup. Let us fix the generator system {f_{1}, ⋯, f_{m}}. We set f_{w} := f_{w_{1}} ∘⋯∘f_{w_{k}} for any w = (w_{1}, ⋯, w_{k}) ∈ {1, ⋯, m}^{k}. We set W = ∪_{n∈ℕ}{1, ⋯, m}^{n} ⋃ Σ_{m} and set W^{∗} = ∪_{n∈ℕ}{1, ⋯, m}^{n}. For any w = (w_{1}, w_{2}, ⋯) ∈ W, we set |w| = n if w ∈ {1, ⋯, m}^{n} and |w| = ∞ if w ∈ Σ_{m}. Furthermore, we set w|k := (w_{1}, ⋯, w_{k}), for any k ∈ ℕ with k ≤ |w|. Moreover, for any w ∈ W^{∗}, we set Σ_{m}(w) := {w′ ∈ Σ_{m} | w′_{j} = w_{j}, j = 1, ⋯, |w|}. For any w^{1} ∈ W^{∗} and w^{2} ∈ W, we set w^{1}w^{2} = (w_{1}^{1}, ⋯, w_{|w^{1}|}^{1}, w_{1}^{2}, w_{2}^{2}, ⋯) ∈ W.

Notation: Let (X, d) be a metric space. For any subset A of X, we set diam A := sup{d(x, y) | x, y ∈ A}. Let μ be a Borel measure on X. We use supp μ to denote the support of μ. For any Borel set A in X, we use μ|_{A} to denote the measure on A such that μ|_{A}(B) = μ(B) for each Borel subset B of A. We set L^{1}(μ) = {φ : X → ℝ | ∫_{X} |φ|dμ < ∞}, with L^{1} norm. For any φ ∈ L^{1}(μ), we sometimes use μ(φ) to mean ∫_{X} φ dμ. For any φ ∈ L^{1}(μ), we use φμ to denote the measure such that (φμ)(A) = ∫_{A} φ dμ for any Borel set A. We set C(X) = {φ : X → ℝ | continuous}. (If X is compact, then C(X) is the Banach space with the supremum norm.) For any subset A of X and any r > 0, we set B(A, r) = {y ∈ X | d(y, A) < r}. For any subset A of X, we use int A to denote the interior of A.

Remark 4. In this paper, we always use the spherical metric on \overline{ℂ}. However, we note that conjugating a rational semigroup G by a Möbius transformation, we may assume that J(G) ⊂ ℂ, and then for a neighborhood V of J(G), the identity map i : (V, d_{s}) → (V, d_{e}) is a bi-Lipschitz map, where d_{s} and d_{e} denote the spherical and Euclidean distance, respectively. In what follows, we often use the above implicitly, especially when we need to use the facts in [F] and [Pe].

3 Main Theorem A

In this section, we show Main Theorem A. We investigate the estimate of the upper box and Hausdorff dimensions of Julia sets of expanding semigroups using thermodynamic formalism in ergodic theory. For the notation used in ergodic theory, see [DGS] and [W2].

Definition 3.1. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated rational semigroup. We say that G is an expanding rational semigroup if J(G) ≠ ∅ and the skew product map f̃: Σ_{m} × \overline{ℂ} → Σ_{m} × \overline{ℂ} associated with the generator system {f_{1}, ⋯, f_{m}} is expanding along fibers, i.e., there exists a positive constant C and a constant λ > 1 such that for each n ∈ ℕ,

inf_{z∈J̃(f̃)} ∥(f̃^{n})′(z)∥ ≥ Cλ^{n},

where we use ∥ · ∥ to denote the norm of the derivative with respect to the spherical metric.

Remark 5. By Theorem 2.6, Theorem 2.8, and Remark 4 in [S2], we see that if G = ⟨f_{1}, ⋯, f_{m}⟩ contains an element of degree at least two, each Möbius transformation in G is neither the identity nor an elliptic element, and G is hyperbolic, i.e., the postcritical set P (G) of G, which is defined as:

P (G) := \overline{⋃_{g∈G}\{all critical values of g\}},

is included in F (G), then G is expanding. Conversely, if G = ⟨f_{1}, ⋯, f_{m}⟩ is expanding, then G is hyperbolic and each Möbius transformation in G is loxodromic. Hence, the notion of expandingness does not depend on any choice of a generator system for a finitely generated rational semigroup.

Lemma 3.2. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated expanding rational semigroup. Suppose ♯J(G) ≤ 2. Then, ♯J(G) = 1 and J(G) is a common repelling fixed point of any f_{j}.

Proof. Suppose ♯J(G) = 2 and let J(G) = {z_{1}, z_{2}}. Then, f_{j} is a Möbius transformation, for each j = 1, ⋯, m. Since G is expanding, each f_{j} is loxodromic. We may assume that z_{1} is a repelling fixed point of f_{1}. Then, since f_{1}^{−1}(J(G)) ⊂ J(G), it follows that z_{2} is an attracting fixed point of f_{1}. This is a contradiction, however, since G is expanding. Hence, ♯J(G) = 1.

Definition 3.3. Let G be a rational semigroup and let t be a non-negative number. We say that a Borel probability measure τ on \overline{ℂ} is t-subconformal (for G) if for each g ∈ G and for each Borel measurable set A in \overline{ℂ}, τ(g(A)) ≤ ∫_{A} ∥g′∥^{t}dτ. Moreover, we set

s(G) = inf{t | ∃τ : t-subconformal measure}.

Definition 3.4. Let X be a compact metric space. Let f : X → X be a continuous map:

1. We use h(f) to denote the topological entropy of f (see p83 in [DGS]). We use h_{μ}(f) to denote the metric entropy of f with respect to an invariant Borel probability measure μ (see p60 in [DGS]).

2. Furthermore, let φ : X → ℝ be a continuous function. Then, we use P (f, φ) to denote the pressure for the dynamics of f and the function φ (see p141 in [DGS]). According to a well known fact: the variational principle (see p142 in [DGS]), we have

P (f, φ) = sup{h_{μ}(f) + ∫_{X} φ dμ},

where the supremum is taken over all f-invariant Borel probability measures μ on X. If an invariant probability measure μ attains the supremum in this manner, then μ is called an equilibrium state for (f, φ). For more details on this notation and the variational principle, see [DGS] and [W2].

3. For a real-valued continuous function φ on X and for each n ∈ ℕ, we define a continuous function S_{n}φ on X as (S_{n}φ)(z) = ∑_{j=0}^{n−1} φ(f^{j}(z)). Note that P (f^{n}, S_{n}φ) = nP (f, φ)(see Theorem 9.8 in [W2]).

Definition 3.5. Let X be a compact metric space and let f : X → X be a continuous map satisfying the fact that there exists a number k ∈ ℕ such that ♯f^{−1}(z) = k for each z ∈ X. Let φ be a continuous function on X. We define an operator L = L_{φ} on C(X) using

Lψ(z) = ∑_{f(z′)=z} exp(φ(z′))ψ(z′).

This is called the transfer operator for (f, φ). Note that L_{φ}^{n} equals the transfer operator for (f^{n}, S_{n}φ), for each n ∈ ℕ.

Lemma 3.6. Let G = ⟨f_{1}, f_{2}, ⋯ f_{m}⟩ be a finitely generated expanding rational semigroup. Let f̃ : Σ_{m} × \overline{ℂ} → Σ_{m} × \overline{ℂ} be the skew product map associated with {f_{1}, ⋯, f_{m}}. Then, for each Hölder continuous function φ on J̃(f̃), the transfer operator L_{φ} for (f̃|_{J̃(f̃)}, φ) on C(J̃(f̃)) satisfies the fact that there exists a unique probability measure ν̃ = ν̃_{φ} on J̃(f̃) satisfying all of the following:

1. L_{φ}^{∗}ν̃ = exp(P)ν̃, where P = P (f̃|_{J̃(f̃)}, φ) is the pressure of (f̃|_{J̃(f̃)}, φ).

2. For each ψ ∈ C(J̃(f̃)), ∥\frac{1}{(exp(P))^{n}} L_{φ}^{n}ψ − ν̃(ψ)α_{φ}∥_{J̃(f̃)} → 0, n → ∞, where we set α_{φ} = lim_{l→∞} \frac{1}{(exp(P))^{l}} L_{φ}^{l}(1) ∈ C(J̃(f̃)) and we use ∥ · ∥_{J̃(f̃)} to

denote the supremum norm on J̃(f̃).

3. α_{φ}ν̃ is f̃-invariant, exact (hence ergodic) and is an equilibrium state for (f̃|_{J̃(f̃)}, φ).

4. α_{φ}(z) > 0 for each z ∈ J̃(f̃).

Proof. According to the Koebe distortion theorem and since the dynamics of σ : Σ_{m} → Σ_{m} is expanding, there exists a number s ∈ ℕ such that the map f̃^{s} : J̃(f̃) → J̃(f̃) satisfies condition I on page 123 in [W1] (each of X_{0}, X, and \overline{X} in [W1] corresponds to J̃(f̃)). Furthermore, by Proposition 3.2 (f) in [S5] and Lemma 3.2, we have the fact that f̃^{s} on J̃(f̃) satisfies condition II on page 125 in [W1]. The map μ → L_{φ}^{∗}μ/(L_{φ}^{∗}μ)(1) is continuous on the space M(J̃(f̃)) of Borel probability measures on J̃(f̃). Hence, this map has a fixed point ν̃ based on the Schauder-Tychonoff fixed point theorem. Let λ = (L_{φ}^{∗}ν̃)(1). Then, L_{φ}^{∗}ν̃ = λν̃. Hence, we have (L_{φ}^{s})^{∗}ν̃ = λ^{s}ν̃. By Theorem 8, Corollary 12, and the statement on equilibrium states on page 140 in [W1], we get λ^{s} = exp(P (f̃^{s}|_{J̃(f̃)}, S_{s}φ)) = exp(sP (f̃|_{J̃(f̃)}, φ)). Hence, we obtain λ = exp(P). The other results also follow from Theorem 8, Corollary 12, and the statement on equilibrium states on page 140 in [W1].

Notation: Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated rational semigroup. Let f̃: Σ_{m} × \overline{ℂ} → Σ_{m} × \overline{ℂ} be the skew product map associated with \{f_{1}, ⋯, f_{m}\}. Suppose that no critical point of f̃ exists in J̃(f̃). Then, we define a function φ̃ on J̃(f̃) as: φ̃((w, x)) := − log ∥(f_{w_{1}})′(x)∥ for (w, x) = ((w_{1}, w_{2}, ⋯), x) ∈ J̃(f̃).

Lemma 3.7. Let G = ⟨f_{1}, f_{2}, ⋯ f_{m}⟩ be a finitely generated expanding rational semigroup. Then, using the above notation, we have the following:

1. The function P (t) = P (f̃|_{J̃(f̃)}, tφ̃) on ℝ is convex and strictly decreasing as t increases. Furthermore, P (t) → −∞ as t → ∞.

2. There exists a unique zero δ ≥ 0 of P (t). Furthermore, if h(f̃|_{J̃(f̃)}) > 0 then δ > 0.

3. There exists a unique probability measure ν̃ = ν̃_{δφ̃} on J̃(f̃) such that M_{δ}^{∗}ν̃ = ν̃, where M_{δ} is an operator on C(J̃(f̃)) defined by

M_{δ}ψ((w, x)) = ∑_{f̃((w′,y))=(w,x)} \frac{ψ((w′, y))}{∥(f_{w′_{1}})′(y)∥^{δ}} .  (1)

Note that M_{δ} = L_{δφ̃}.

4. δ satisfies the fact that

δ = \frac{h_{αν̃}(f̃)}{− ∫_{J̃(f̃)} φ̃αdν̃} ≤ \frac{log(∑_{j=1}^{m} deg(f_{j}))}{− ∫_{J̃(f̃)} φ̃αdν̃},  (2)

where α = lim_{l→∞} M_{δ}^{l}(1) ∈ C(J̃(f̃)).

Proof. Using the variational principle, we have P (t) = sup{h_{μ}(f̃|_{J̃(f̃)})+∫_{J̃(f̃)} tφ̃ dμ},

where the supremum is taken over all f̃-invariant Borel probability measures μ on J̃(f̃). In addition, note that by Theorem 6.1 in [S5], we determine that the topological entropy h(f̃) of f̃ on Σ_{m} × \overline{ℂ} is less than or equal to log(∑_{j=1}^{m} deg(f_{j})). By the variational principle: h(f̃) = sup{h_{μ}(f̃) | f̃_{∗}μ = μ} (see p138 in [DGS] or Theorem 8.6 in [W2]); it follows that

h_{μ}(f̃) ≤ log(∑_{j=1}^{m}(deg(f_{j})))

for any f̃-invariant Borel probability measure μ on J̃(f̃). Combining this with the fact that the dynamics of f̃ on J̃(f̃) is expanding, we see that the function P (t) on ℝ is convex, strictly decreasing as t increases, and P (t) → −∞ as t → ∞. Hence, there exists a unique number δ ∈ ℝ satisfying P (δ) = 0. Since P (0) = h(f̃|_{J̃(f̃)}), we have δ > 0 if h(f̃|_{J̃(f̃)}) > 0. The statements 3 and 4 follow from Lemma 3.6 and this argument.

Definition 3.8. We define an operator M̂_{δ} acting on the space of all Borel measurable functions on J̃(f̃) using the same formula as that for M_{δ}. (See (1)).

We now show that M̂_{δ} acts on L^{1}(ν̃) and that M̂_{δ} on L^{1}(ν̃) is a bounded operator, where ν̃ = ν̃_{δφ̃}.

Lemma 3.9. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated expanding rational semigroup. Using the above notation, we have the following:

1. Let A be a Borel set in J̃(f̃). If ν̃(A) = 0, then ν̃(f̃^{−1}(A)) = 0.

2. Let ψ be a Borel measurable function on J̃(f̃). Let {ψ_{n}}_{n} be a sequence of Borel measurable functions on J̃(f̃). Suppose ψ_{n}(z) → ψ(z) for almost every z ∈ J̃(f̃) with respect to ν̃. Then, we have (M̂_{δ}ψ_{n})(z) → (M̂_{δ}ψ)(z) for almost every z ∈ J̃(f̃) with respect to ν̃.

3. If ψ ∈ L^{1}(ν̃), then M̂_{δ}ψ ∈ L^{1}(ν̃). Furthermore, M̂_{δ} is a bounded operator on L^{1}(ν̃) and the operator norm ∥M̂_{δ}∥ is equal to 1.

Proof. Let μ = αν̃, where α = lim_{l→∞} M_{δ}^{l}1. Then, by Lemma 3.6-3, we have f̃_{∗}μ = μ. Furthermore, by Lemma 3.6-4, μ and ν̃ are absolutely continuous with respect to each other. Hence, we obtain the statement 1, and the statement 2 follows easily from this.

We now show the statement 3. First, we show the following claim: Claim: for any ψ ∈ C(J̃(f̃)), we have ∫ |M̂_{δ}ψ| dν̃ ≤ ∫ |ψ| dν̃.

To show this claim, let ψ ∈ C(J̃(f̃)). Let ψ^{+} = max{ψ, 0} and ψ^{−} = − min{ψ, 0}. Then, we have ψ = ψ^{+} − ψ^{−} and |ψ| = ψ^{+} + ψ^{−}. Since M_{δ}^{∗}ν̃ = ν̃, we obtain ∫ |M̂_{δ}ψ| dν̃ = ∫ |M_{δ}ψ^{+} − M_{δ}ψ^{−}| dν̃ ≤ ∫ M_{δ}ψ^{+} dν̃ + ∫ M_{δ}ψ^{−} dν̃ = ∫ ψ^{+} + ψ^{−} dν̃ = ∫ |ψ| dν̃. Hence, the above claim holds.

Now, let ψ be a general element of L^{1}(ν̃). Let {ψ_{n}}_{n} be a sequence in C(J̃(f̃)) such that ψ_{n} → ψ in L^{1}(ν̃). We may assume that ψ_{n}(z) → ψ(z) for almost every z ∈ J̃(f̃) with respect to ν̃. Then, according to the statement 2, we have (M̂_{δ}ψ_{n})(z) → (M̂_{δ}ψ)(z) for almost every z ∈ J̃(f̃) with respect to ν̃. Using this claim, {M̂_{δ}ψ_{n}}_{n} is a Cauchy sequence in L^{1}(ν̃). Hence, it follows that M̂_{δ}ψ ∈ L^{1}(ν̃). Furthermore, we have ∫ |M̂_{δ}ψ| dν̃ = lim_{n→∞} ∫ |M̂_{δ}ψ_{n}| dν̃ ≤ lim_{n→∞} ∫ |ψ_{n}| dν̃ = ∫ |ψ| dν̃. Hence, ∥M̂_{δ}∥ ≤ 1. Since ∫ M̂_{δ}1 dν̃ = ∫ 1 dν̃ = 1, we obtain ∥M̂_{δ}∥ = 1.

We now show that the measure ν̃ = ν̃_{δφ̃} is “conformal”.

Lemma 3.10. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated expanding rational semigroup. Let k ∈ ℕ and let A be a Borel set in J̃(f̃) such that f̃^{k} : A → f̃^{k}(A) is injective. Then, using the above notation, we have ν̃(f̃^{k}(A)) = ∫_{A} ∥(f̃^{k})′∥^{δ} dν̃.

Proof. We have M_{δ}^{k}ν̃ = ν̃ and M_{δ}^{k} is a transfer operator for (f̃^{k}, δS_{k}φ̃). By Proposition 2.2 in [DU] and Lemma 3.9-3, we obtain the statement.

Lemma 3.11. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated expanding rational semigroup. Then, with our notation, the probability measure ν := (π\overline{ℂ})_{∗}(ν̃) is δ-subconformal.

Proof. First, note that by Lemma 3.10, it follows that for any Borel set B in Σ_{m}×\overline{ℂ} we have ν̃(f̃^{k}(B)) ≤ ∑_{j} ν̃(f̃^{k}(B_{j})) = ∑_{j} ∫_{B_{j}} ∥(f̃^{k})′∥^{δ} dν̃ = ∫_{B} ∥(f̃^{k})′∥^{δ} dν̃,

where B = ∑_{j} B_{j} is a measurable partition such that f̃^{k}|_{B_{j}} is injective for each j. Hence, for any Borel set A in \overline{ℂ} and any w ∈ W^{∗} with |w| = k, it follows

that ν(f_{w}(A)) = ν̃(π−1 ; \overline{ℂ}(f_{w}(A))) = ν̃(f̃^{k}(Σ_{m}(w)×A)) ≤ ∫_{Σ_{m}(w)×A} ∥(f̃^{k})′∥^{δ}dν̃ ≤

∫_{A} ∥(f_{w})′∥^{δ}dν.

We now consider the Poincaré series and critical exponent for a rational semigroup.

Definition 3.12. Let G be a rational semigroup. We set

A(G) = \overline{∪_{g∈G}g(\{z ∈ \overline{ℂ} | ∃h ∈ G, h(z) = z, |h′(z)| < 1\})}.

For any s ≥ 0 and x ∈ \overline{ℂ}, we set S(s, x) = ∑_{g∈G} ∑_{g(y)=x} ∥g′(y)∥^{−s}. Furthermore, we set S(x) = inf{s ≥ 0 | S(s, x) < ∞} (If no s exists with S(s, x) < ∞, then we set S(x) = ∞). We set s_{0}(G) = inf\{S(x) | x ∈ \overline{ℂ}\}.

If G is generated by finite elements \{f_{1}, ⋯, f_{m}\}, then for any x ∈ \overline{ℂ} and t ≥ 0, we set T (t, x) = ∑_{w∈W^{∗}} ∑_{f_{w}(y)=x} ∥(f_{w})′(y)∥^{−t} and T (x) = inf{t ≥ 0 | T (t, x) < ∞} (If no t exists with T (t, x) < ∞, then we set T (x) = ∞). Note that S(x) ≤ T (x).

Lemma 3.13. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated rational semigroup. Let f̃: Σ_{m} × \overline{ℂ} → Σ_{m} × \overline{ℂ} be a skew product map associated with \{f_{1}, ⋯, f_{m}\}. Let z ∈ F̃(f̃) be a point. Then, there exists a number n ∈ ℕ such that π\overline{ℂ}(f̃^{n}(z)) ∈ F (G).

Proof. Let z ∈ F̃(f̃) be a point. Then, there exists a word w ∈ W^{∗} and an open neighborhood V of π\overline{ℂ}(z) in \overline{ℂ} such that z ∈ Σ_{m}(w) × V ⊂ F̃(f̃). Let n = |w|. Then, F̃(f̃) ⊃ f̃^{n}(Σ_{m}(w) × V) = Σ_{m} × f_{w}(V). Since π\overline{ℂ}J̃(f̃) = J(G) (Proposition 3.2 in [S5]), it follows that f_{w}(V) ⊂ F (G). Hence, π\overline{ℂ}f̃^{n}(z) = f_{w}(π\overline{ℂ}(z)) ∈ f_{w}(V) ⊂ F (G).

Lemma 3.14. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated expanding rational semigroup. Let f̃: Σ_{m} × \overline{ℂ} → Σ_{m} × \overline{ℂ} be a skew product map associated with \{f_{1}, ⋯, f_{m}\}. Let z ∈ Σ_{m} × \overline{ℂ} be a point with x := π\overline{ℂ}(z) ∈ \overline{ℂ} \ A(G). Then, for each open neighborhood V of J̃(f̃) in Σ_{m} × \overline{ℂ}, there exists a number l ∈ ℕ such that ∪_{n≥l}(f̃^{n})^{−1}(z) ⊂ V. Furthermore, we have A(G) ∪ P (G) ⊂ F (G) and if x ∈ \overline{ℂ} \ (A(G) ∪ P (G)), then T (x) < ∞.

Proof. By Remark 5, we have P (G) ⊂ F (G). Next we show A(G) ⊂ F (G). Since G is expanding, then using the Koebe distortion theorem and π\overline{ℂ}(J̃(f̃)) = J(G) (Proposition 3.2 in [S5]), we obtain that there exist an n ∈ ℕ and a number ε > 0 such that for each a ∈ J(G) and each w ∈ W^{∗} with |w| = n, we can take well-defined inverse branches of f_{w}^{−1} on B(a, ε) and any inverse branch γ of f_{w}^{−1} on B(a, ε) satisfies γ(B(a, ε)) ⊂ B(γ(a), \frac{1}{2}ε) and ∥γ′(y)∥ ≤ \frac{1}{2} for each y ∈ B(a, ε). Taking a small enough ε, it follows that for each a ∈ J(G) and each w ∈ W^{∗}, we can take well-defined inverse branches γ of f_{w}^{−1} on B(a, ε) and we have

sup{∥γ′(y)∥ | y ∈ B(a, ε), a ∈ J(G), γ : a branch of f_{w}^{−1}, |w| = n} → 0

as n → ∞. Let y ∈ \overline{ℂ} be a point such that g(y) = y and |g′(y)| < 1 for some g ∈ G. Suppose that there exist an element h ∈ G and a point a ∈ J(G) such that h(y) ∈ B(a, ε). Let γ_{n} be a well-defined inverse branch of (hg^{n})^{−1} on B(a, ε) such that γ_{n}(hg^{n}(y)) = γ_{n}(h(y)) = y. Then |γ′_{n}(y)| → ∞ as n → ∞. This contradicts the previous argument. Hence A(G) ⊂ \overline{ℂ}\B(J(G), ε) ⊂ F (G).

Next, suppose that there exists a sequence (z_{j}) in F̃(f̃) such that f̃^{n_{j}}(z_{j}) = z and z_{j} → z_{∞} ∈ F̃(f̃) where n_{j} ∈ ℕ with n_{j} → ∞ as j → ∞. Then, by Lemma 3.13, there exists a number n ∈ ℕ such that π\overline{ℂ}(f̃^{n}(z_{∞})) ∈ F (G). Let x_{j} = π\overline{ℂ}(f̃^{n}(z_{j})) for each j ∈ ℕ and let x_{∞} = π\overline{ℂ}f̃^{n}(z_{∞}). Then, for each j with n_{j} > n, there exists an element g_{j} ∈ G such that g_{j}(x_{j}) = x. Let α = d(x, A(G)). Since x_{j} → x_{∞} ∈ F (G), we have ♯\{j | d(g_{j}(x_{∞}), x) < \frac{α}{2}\} = ∞.

By contrast, we have sup{d(f_{w}(x_{∞}), A(G)) | |w| = n} → 0 as n → ∞. For, if P (G) ≠ ∅, the above follows from Theorem 1.34 in [S3]. Even if P (G) = ∅, since G is expanding, by the Koebe distortion theorem, then for each z ∈ F (G), \overline{∪_{g∈G}g(z)} ⊂ F (G). Using the same argument as in the proof of Theorem 1.34 in [S3], we obtain the above.

Hence, we obtain a contradiction. Therefore, we have shown that for each open neighborhood V of J̃(f̃) in Σ_{m} × \overline{ℂ}, there exists a number l ∈ ℕ such that ∪_{n≥l}(f̃^{n})^{−1}(z) ⊂ V. If x ∈ \overline{ℂ} \ (A(G) ∪ P (G)), then since G is expanding, combining the above result with the Koebe distortion theorem, we obtain T (x) < ∞.

Definition 3.15. Let E be a subset of \overline{ℂ}, t ≥ 0 a number and β > 0 a number. We set

H_{β}^{t}(E) := inf{∑_{i=1}^{∞}(diam(U_{i}))^{t} | diam(U_{i}) ≤ β, E ⊂ ∪_{i=1}^{∞}U_{i}}

and H^{t}(E) = lim_{β→0} H_{β}^{t}(E) with respect to the spherical metric on \overline{ℂ}. H^{t}(E) is called the t-dimensional (outer) Hausdorff measure of E with respect to the spherical metric. Note that H^{t}(E) is a Borel regular measure on \overline{ℂ}(see [R]). We set dim_{H}(E) := sup{t ≥ 0 | H^{t}(E) = ∞} = inf{t ≥ 0 | H^{t}(E) = 0}. dim_{H}(E) is called the Hausdorff dimension of E. Furthermore, let N_{r}(E) be the smallest number of sets of spherical diameter r that can cover E. We set \overline{dim}_{B}(E) = lim sup_{r→0} \frac{log N_{r}(E)}{− log r}. \overline{dim}_{B}(E) is called the upper box dimension of E.

Lemma 3.16. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated expanding rational semigroup. Let τ be a t-subconformal measure. Then, there exists a positive constant c such that for each r with 0 < r < diam \overline{ℂ} and each x ∈ J(G), we have τ(B(x, r)) ≥ cr^{t}. Furthermore, H^{t}|_{J(G)} is absolutely continuous with respect to τ, H^{t}(J(G)) < ∞ and \overline{dim}_{B}(J(G)) ≤ t.

Proof. Let τ be a t-subconformal measure. Using the argument in the proof of Theorem 3.4 in [S2], we find that there exists a positive constant c such that for each r with 0 < r < diam \overline{ℂ} and each x ∈ J(G), τ(B(x, r)) ≥ cr^{t}. (Note that for an estimate of this type, we need only expandingness and we do not need the strong open set condition used in [S2].) By Proposition 2.2 in [F], we find that H^{t}|_{J(G)} is absolutely continuous with respect to τ. In particular, H^{t}(J(G)) < ∞. Furthermore, by Theorem 7.1 in [Pe], we get \overline{dim}_{B}(J(G)) ≤ t.

Using these arguments, we now demonstrate Main Theorem A.

Proof of Main Theorem A: By Lemma 3.7, we have that the function P (t) has a unique zero δ, there exists a unique probability measure ν̃ on J̃(f) such that M_{δ}^{∗}ν̃ = ν̃, and δ satisfies (2).

By Lemma 3.16, \overline{dim}_{B}(J(G)) ≤ s(G). By Lemma 3.14, we have A(G) ∪ P (G) ⊂ F (G). Let x ∈ \overline{ℂ} \ (A(G) ∪ P (G)) be a point. According to Theorem 4.2 in [S2] and the fact that S(x) ≤ T (x) < ∞ (Lemma 3.14), we obtain s(G) ≤ s_{0}(G) ≤ S(x) ≤ T (x).

We show δ = T (x). We consider the following two cases: Case 1: T (T (x), x) = ∞. Case 2: T (T (x), x) < ∞. Suppose we have Case 1. Let z ∈ Σ_{m} × \overline{ℂ} be a point with π\overline{ℂ}(z) = x. Let t_{n} be a sequence of real numbers such that t_{n} > T (x) for each n ∈ ℕ and t_{n} → T (x). For each n ∈ ℕ, let μ_{n} be a Borel probability measure on Σ_{m} × \overline{ℂ} defined by:

μ_{n} = \frac{1}{T (t_{n}, x)} ∑_{p∈ℕ} ∑_{f̃^{p}(z′)=z} ∥(f̃^{p})′(z′)∥^{−t_{n}}δ_{z′},

where δ_{z′} denotes the Dirac measure concentrated at z′. Since the space of Borel probability measures on Σ_{m} × \overline{ℂ} is compact, we may assume that there exists a Borel probability measure μ_{∞} on Σ_{m} × \overline{ℂ} such that μ_{n} → μ_{∞} as n → ∞, with respect to the weak topology. Then, by Lemma 3.14, we have supp μ_{∞} ⊂ J̃(f̃). We now show the following claim: Claim 1: For any Borel set A in J̃(f̃) such that f̃: A → f̃(A) is injective, we have μ_{∞}(f̃(A)) = ∫_{A} ∥(f̃)′∥^{T (x)} dμ_{∞}.

To show this claim, let A be a Borel set in Σ_{m} × \overline{ℂ} such that f̃: A → f̃(A) is injective. Then, μ_{n}(f̃(A)) = ∫_{A} ∥(f̃)′∥^{t_{n}} dμ_{n} − \frac{1}{T (t_{n},x)}♯(f̃^{−1}(z) ∩ A). If A satisfies that μ_{∞}(∂f̃(A)) = μ_{∞}(∂A) = 0, then letting n → ∞ in the above, it follows that μ_{∞}(f̃(A)) = ∫_{A} ∥(f̃)′∥^{T (x)} dμ_{∞}.

Now let B be a general Borel set in J̃(f̃) such that f̃: B → f̃(B) is injective. Then, let B = ∑_{j∈ℕ} B_{j} be a countable disjoint union of Borel sets B_{j} satisfying the fact that for each j ∈ ℕ, there exists an open neighborhood W_{j} of \overline{B_{j}} in Σ_{m} × \overline{ℂ} such that f̃ : W_{j} → f̃(W_{j}) is a homeomorphism. Let j be a fixed number and K a fixed compact subset of W_{j}. Then, for each n ∈ ℕ, there exists a number ϵ_{n} > 0 such that the set V_{n} := \{z ∈ Σ_{m} × \overline{ℂ} | d(z, K) < ϵ_{n}\} satisfies V_{n} ⊂ W_{j}, μ_{∞}(∂V_{n}) = μ_{∞}(∂(f̃(V_{n}))) = 0, μ_{∞}(f̃(V_{n}) \ f̃(K)) < \frac{1}{n} and μ_{∞}(V_{n} \ K) < \frac{1}{n}. For these sets V_{n}, by the previous argument, we have μ_{∞}(f̃(V_{n})) = ∫_{V_{n}} ∥(f̃)′∥^{T (x)} dμ_{∞}. Letting n → ∞, we obtain μ_{∞}(f̃(K)) = ∫_{K} ∥(f̃)′∥^{T (x)} dμ_{∞}. Next, for each l ∈ ℕ, we can take a compact subset K_{l} of B_{j} such that μ_{∞}(B_{j} \ K_{l}) < \frac{1}{l} and μ_{∞}(f̃(B_{j}) \ f̃(K_{l})) < \frac{1}{l}. For these sets K_{l}, using the above argument, we have μ_{∞}(f̃(K_{l})) = ∫_{K_{l}} ∥(f̃)′∥^{T (x)} dμ_{∞}. Letting l → ∞, we obtain μ_{∞}(f̃(B_{j})) = ∫_{B_{j}} ∥(f̃)′∥^{T (x)} dμ_{∞}. Since B = ∑_{j} B_{j} and f̃ is injective on B, we obtain μ_{∞}(f̃(B)) = ∫_{B} ∥(f̃)′∥^{T (x)} dμ_{∞}. Hence, we have shown Claim 1.

Using Claim 1 and Proposition 2.2 in [DU], it follows that L_{T (x)φ̃}^{∗}μ_{∞} = μ_{∞}. We now show that δ = T (x). Suppose δ < T (x). Then, by Lemma 3.7-

1, we have P (T (x)) < 0. Then, for each ψ ∈ C(J̃(f̃)), we have μ_{∞}(ψ) = (exp P (T (x)))^{l} · μ_{∞}(\frac{L_{T (x)φ̃}ψ}{(exp(P (T (x))))^{l}}) → 0 as l → ∞, by Lemma 3.6-2. Hence, μ_{∞}(ψ) = 0 and this implies a contradiction. Suppose T (x) < δ. Then, by a similar argument to the one above, we get a contradiction. Hence, T (x) = δ.

We now consider Case 2: T (T (x), x) < ∞. Let z ∈ Σ_{m} × \overline{ℂ} be a point with x = π\overline{ℂ}(z). Then, we take Patterson’s function([Pa]) Φ: i.e., Φ is a continuous, non-decreasing function from ℝ_{+} := {t ∈ ℝ | t ≥ 0} to ℝ_{+} that satisfies the following:

1. Q(t) := ∑_{n} ∑_{f̃^{n}(z′)=z} Φ(∥(f̃^{n})′(z′)∥)∥(f̃^{n})′(z′)∥^{−t} converges for each t > T (x) and does not converge for each t ≤ T (x).

2. For each ϵ > 0, there is a number r_{0} ∈ ℝ_{+} such that Φ(rs) ≤ s^{ϵ}Φ(r) for each r > r_{0} and each s > 1.

Let t_{n} be a sequence of ℝ such that t_{n} > T (x) for each n ∈ ℕ, t_{n} → T (x) as n → ∞ and the measures:

τ_{n} := \frac{1}{Q(t_{n})} ∑_{p} ∑_{f̃^{p}(z′)=z} Φ(∥(f̃^{p})′(z′)∥)∥(f̃^{p})′(z′)∥^{−t_{n}}δ_{z′}

tend to a Borel probability measure τ_{∞} on Σ_{m} × \overline{ℂ} as n → ∞. Then, by Lemma 3.14, we have supp τ_{∞} ⊂ J̃(f̃). Furthermore, combining the argument in the proof of Claim 1 in Case 1 with the properties of Φ, we find that for each Borel set A in J̃(f̃) such that f̃ : A → f̃(A) is injective, τ_{∞}(f̃(A)) = ∫_{A} ∥(f̃)′∥^{T (x)} dτ_{∞}. Combining this with the argument used in Case 1, we obtain δ = T (x).

Since G is expanding and ν is δ-subconformal (Lemma 3.11), using an argument in the proof of Theorem 4.4 in [S2], we obtain supp ν ⊃ J(G). Hence, supp ν = J(G).

Hence, we have shown Main Theorem A.

Corollary 3.17. Let G = ⟨f_{1}, f_{2}, ⋯ f_{m}⟩ be a finitely generated expanding rational semigroup. Then, \overline{dim}_{B}(J(G)) ≤ δ ≤ ^{\frac{log(∑_{j=1}^{m} deg(f_{j}))}{log λ}}, where λ denotes the number in Definition 3.1.

Proof. By Main Theorem A and (2), we have

\overline{dim}_{B}(J(G)) ≤ δ ≤ \frac{log(∑_{j=1}^{m} deg(f_{j}))}{− ∫_{J̃(f̃)} φ̃αdν̃}

= \frac{n log(∑_{j=1}^{m} deg(f_{j}))}{− ∫_{J̃(f̃)} S_{n}φ̃ αdν̃}

≤ \frac{n log(∑_{j=1}^{m} deg(f_{j}))}{log C + n log λ},

for each n ∈ ℕ. Letting n → ∞, we obtain the result.

4 Conformal measure

In this section we introduce the notion of “conformal measure”, which is needed in Main Theorem B.

Definition 4.1. 1. Let G be a rational semigroup. Let t ∈ ℝ with t ≥ 0. We say that a Borel probability measure τ on J(G) is t-conformal (for G) if for any Borel set A and g ∈ G, if A, g(A) ⊂ J(G) and g : A → g(A) is injective, then

τ(g(A)) = ∫_{A} ∥g′∥^{t} dτ.

2. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated rational semigroup. We say that a Borel probability measure μ on J(G) satisfies the separating condition for {f_{1}, ⋯, f_{m}} if μ(f_{i}^{−1}(J(G)) ∩ f_{j}^{−1}(J(G))) = 0 for any (i, j) with i, j ∈ {1, ⋯, m} and i ≠ j.

We show some fundamental properties of conformal measures.

Lemma 4.2. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated rational semigroup. Let τ be a t-conformal measure. Then, τ is a t-subconformal measure.

Proof. Let A be a Borel set in \overline{ℂ} and g an element of G. Let J(G) = ∑ B_{i} be a measurable partition of J(G) such that we can take the well-defined inverse branches of g^{−1} on B_{i}, for each i (we divide J(G) into {B_{i}} so that for a critical value c ∈ J(G) of g, there exists an i such that B_{i} = {c}). Let {C_{i,j}}_{j} be the images of B_{i} using the inverse branches of g^{−1} so that g : C_{i,j} → B_{i} is bijective for each j. Then, we have τ(g(A)) = τ(g(A) ∩ J(G)) = ∑_{i} τ(g(A) ∩ B_{i}) ≤ ∑_{i,j} τ(g(A ∩ C_{i,j})) = ∑_{i,j} ∫_{A∩C_{i,j}} ∥g′∥^{t}dτ = ∫_{A∩∪_{i,j}C_{i,j}} ∥g′∥^{t}dτ ≤ ∫_{A} ∥g′∥^{t}dτ. Hence, τ is t-subconformal.

Lemma 4.3. Let G be a rational semigroup. Let τ be a Borel probability measure on J(G), g ∈ G an element, and V an open set in \overline{ℂ} with V ∩g^{−1}(J(G)) ≠ ∅. Suppose that g : V → g(V) is a homeomorphism and that for any Borel set A in V ∩ g^{−1}(J(G)), τ(g(A)) = ∫_{A} ∥g′∥^{t}dτ. Let h := (g|_{V})^{−1} : g(V) → V. Then, we find that for any Borel set B in g(V) ∩ J(G), τ(h(B)) = ∫_{B} ∥h′∥^{t}dτ.

Proof. Let μ := h_{∗}(τ|_{g(V)∩J(G)}). Then, by the assumption, dμ = ∥g′∥^{t}dτ′, where τ′ = τ|_{V ∩g^{−1}J(G)}. Let B be a Borel set in g(V) ∩ J(G). Then, τ(h(B)) = ∫_{h(B)} ∥g′∥^{−t} · ∥g′∥^{t}dτ = ∫_{h(B)} ∥g′∥^{−t}dμ = ∫_{B} ∥g′∥^{−t} ∘ hdτ = ∫_{B} ∥h′∥^{t}dτ.

Lemma 4.4. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated rational semigroup. Let f̃: Σ_{m} ×\overline{ℂ} → Σ_{m} ×\overline{ℂ} be the skew product map associated with \{f_{1}, ⋯, f_{m}\}. Let τ̃ be a Borel probability measure on J̃(f̃), n ∈ ℕ an integer, and V an open set in Σ_{m} × \overline{ℂ} such that V ∩ J̃(f̃) ≠ ∅. Suppose that f̃^{n} : V → f̃^{n}(V) is a homeomorphism and that for any Borel set A in J̃(f̃), τ̃(f̃^{n}(A)) = ∫_{A} ∥(f̃^{n})′∥^{t} dτ̃. Let h = (f̃^{n}|_{V})^{−1} : f̃^{n}(V) → V. Then, we obtain the result that for any Borel set B in f̃^{n}(V) ∩ J̃(f̃), τ̃(h(B)) = ∫_{B} ∥(f̃^{n})′(h)∥^{−t} dτ̃.

Proof. This lemma can be shown using the same method as in the proof of Lemma 4.3.

Lemma 4.5. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated rational semigroup. Let τ be a t-conformal measure satisfying the separating condition for {f_{1}, ⋯, f_{m}}. Suppose that for any g ∈ G, if c is a critical point of g with g(c) ∈ J(G), then τ({c}) = 0. Then, for any k ∈ ℕ, τ(f_{w}^{−1}(J(G)) ∩ f_{w′}^{−1}(J(G))) = 0 for any w = (w_{1}, ⋯, w_{k}), w′ = (w′_{1}, ⋯, w′_{k}) ∈ {1, ⋯, m}^{k} with w ≠ w′.

Proof. Let w = (w_{1}, ⋯, w_{k}), w′ = (w′_{1}, ⋯, w′_{k}) ∈ {1, ⋯, m}^{k} with w ≠ w′. Let 1 ≤ u ≤ k be the maximum such that w_{u} ≠ w′_{u}. If u = k, then τ(f_{w}^{−1}(J(G)) ∩ f_{w′}^{−1}(J(G))) ≤ τ(f_{w_{k}}^{−1}(J(G)) ∩ f_{w′_{k}}^{−1}(J(G))) = 0. Suppose that u < k. Let g = f_{w_{u+1}} ⋯ f_{w_{k}} = f_{w′_{u+1}} ⋯ f_{w′_{k}}. Then, f_{w}^{−1}(J(G)) ∩ f_{w′}^{−1}(J(G)) ⊂ g^{−1}(f_{w_{u}}^{−1}(J(G)) ∩ f_{w′_{u}}^{−1}(J(G))). By Lemma 4.3, we have τ(g^{−1}(f_{w_{u}}^{−1}(J(G)) ∩ f_{w′_{u}}^{−1}(J(G)))) = 0. Hence, we obtain τ(f_{w}^{−1}(J(G)) ∩ f_{w′}^{−1}(J(G))) = 0.

Definition 4.6. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a rational semigroup. Suppose that for each g ∈ G, no critical value of g exists in J(G). Let t ∈ ℝ. We define an operator N_{t} : C(J(G)) → C(J(G)) as follows: (N_{t}ψ)(z) = ∑_{j=1}^{m} ∑_{f_{j}(y)=z} ∥f′_{j}(y)∥^{−t}ψ(y) for each ψ ∈ C(J(G)).

Lemma 4.7. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a rational semigroup. Suppose that for each g ∈ G, no critical value of g exists in J(G). Let f̃: Σ_{m} × \overline{ℂ} → Σ_{m} × \overline{ℂ} be the skew product map associated with {f_{1}, ⋯, f_{m}}. Then, we have the following commutative diagram:

C(J(G)) ⟶^{N_{t}} C(J(G))
(π\overline{ℂ})^{∗}↓  ↓^{(π\overline{ℂ})^{∗}}
C(J̃(f̃)) ⟶_{L_{tφ̃}} C(J̃(f̃)).

Proof. Let ψ ∈ C(J̃(f̃)) and (w, x) ∈ J̃(f̃). Then, ((π\overline{ℂ})^{∗}N_{t}ψ)((w, x)) = (N_{t}ψ)(x) = ∑_{j=1}^{m} ∑_{f_{j}(y)=x} ∥f′_{j}(y)∥^{−t}ψ(y). Conversely, (L_{tφ̃}(π\overline{ℂ})^{∗}ψ)((w, x)) = ∑_{f̃((w′,y))=(w,x)} ∥f′_{w′_{1}}(y)∥^{−t}((π\overline{ℂ})^{∗}ψ)((w′, y)) = ∑_{j=1}^{m} ∑_{f_{j}(y)=x} ∥f′_{j}(y)∥^{−t}ψ(y).

Lemma 4.8. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a rational semigroup. Suppose that for each g ∈ G, no critical value of g exists in J(G). Then, we have the following:

1. Let τ be a t-conformal measure. Then, we have N_{t}^{∗}τ ≥ τ; i.e., for each ψ ∈ C(J(G)) such that 0 ≤ ψ(z) for each z ∈ J(G), we have (N_{t}^{∗}τ)(ψ) ≥ τ(ψ).

2. If τ is a t-conformal measure satisfying the separating condition for {f_{1}, ⋯, f_{m}}, then N_{t}^{∗}τ = τ.

3. If τ is a t-conformal measure satisfying N_{t}^{∗}τ = τ, then τ satisfies the separating condition for {f_{1}, ⋯, f_{m}}.

Proof. Let J(G) = ∑_{i=1}^{u} B_{i} be a measurable partition of J(G) such that for each j = 1, ⋯, m and i = 1, ⋯, u, we can take the well-defined inverse branches of f_{j}^{−1} on B_{i}. Then, for any Borel probability measure τ on J(G) and any ψ ∈ C(J(G)), we have

∫ N_{t}ψ dτ = ∫ ∑_{j=1}^{m} ∑_{f_{j}(y)=z} ∥f′_{j}(y)∥^{−t}ψ(y) dτ(z)

= ∑_{j} ∑_{i} ∑_{γ} ∫_{B_{i}} ∥f′_{j}(γ(z))∥^{−t}ψ(γ(z)) dτ(z),

where γ runs over all inverse branches of f_{j}^{−1} on B_{i}. Suppose that τ is tconformal. Then, we have

∫_{B_{i}} ∥f′_{j}(γ(z))∥^{−t}ψ(γ(z)) dτ(z) = ∫_{γ(B_{i})} ∥f′_{j}(x)∥^{−t}ψ(x) d(γ_{∗}(τ|_{B_{i}}))(x)
= ∫_{γ(B_{i})} ∥f′_{j}(x)∥^{−t}ψ(x) · ∥f′_{j}(x)∥^{t} dτ(x)
= ∫_{γ(B_{i})} ψ(x) dτ(x).

Hence, ∫ N_{t}ψ dτ = ∑_{j} ∑_{i} ∑_{γ} ∫_{γ(B_{i})} ψ dτ, which is larger than or equal to ∫_{J(G)} ψ dτ if 0 ≤ ψ(z) for each z ∈ J(G), since J(G) = ∪_{j=1}^{m}f_{j}^{−1}(J(G)) (Lemma 1.1.4 in [S1]). Furthermore, if τ is a t-conformal measure satisfying the separating condition for {f_{1}, ⋯, f_{m}}, then for each ψ ∈ C(J(G)), we have ∫ N_{t}ψ dτ = ∑_{j} ∑_{i} ∑_{γ} ∫_{γ(B_{i})} ψ dτ = ∫_{J(G)} ψ dτ, by J(G) = ∪_{j=1}^{m}f_{j}^{−1}(J(G)).

We now show the statement 3. Let τ be a t-conformal measure satisfying N_{t}^{∗}τ = τ. Let ψ ∈ C(J(G)) be an element with ψ(x) ≥ 0 for each x ∈ J(G). Then, by the above argument, it follows that ∫_{J(G)} N_{t}ψ dτ = ∑_{j} ∑_{i} ∑_{γ} ∫_{γ(B_{i})} ψ dτ ≥ ∫_{J(G)} ψ dτ, where γ runs over all inverse branches of f_{j}^{−1} on B_{i}. Since N_{t}^{∗}τ = τ, we have the equality shown above. Hence, τ satisfies the separating condition for {f_{1}, ⋯, f_{m}}.

Lemma 4.9. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated expanding rational semigroup. Let δ be the number in Lemma 3.7, t ≥ 0 a number, and ν̃_{tφ̃} the Borel probability measure on J̃(f̃) that is obtained in Lemma 3.6( the unique fixed point of L_{tφ̃}^{∗}). Let ν_{t} := (π\overline{ℂ})_{∗}ν̃_{tφ̃}. Then, we have the following:

1. ν := ν_{δ} satisfies N_{δ}^{∗}ν = ν.

2. \frac{1}{(exp(P (t)))^{l}} N_{t}^{l}ψ → ν_{t}(ψ)· lim_{l→∞} N_{t}^{l}1 in C(J(G)), where P (t) = P (f̃|_{J̃(f̃)}, tφ̃).

3. If τ is a Borel probability measure on J(G) such that N_{t}^{∗}τ = τ, then t = δ and τ = ν.

Proof. By Lemma 4.7, we obtain the statement 1. Since π\overline{ℂ}(J̃(f̃)) = J(G) (Proposition 3.2 in [S5]), we find that (π\overline{ℂ})^{∗} : C(J(G)) → C(J̃(f̃)) is an isometry with respect to the supremum norms. Hence, by Lemma 3.6 and Lemma 4.7, we find that \{\frac{1}{(exp(P (t)))^{l}} N_{t}^{l}ψ\}_{l∈ℕ} is a Cauchy sequence in C(J(G)). Let ψ_{0} = lim_{l→∞} \frac{1}{(exp(P (t)))^{l}} N_{t}^{l}ψ. Then, by Lemma 3.6, we obtain

(π\overline{ℂ})^{∗}ψ_{0} = lim_{l→∞} \frac{1}{(exp(P (t)))^{l}} L_{tφ̃}^{l}(π\overline{ℂ})^{∗}ψ

= ν̃_{tφ̃}((π\overline{ℂ})^{∗}ψ) · α_{tφ̃}

= ν_{t}(ψ) · lim_{l→∞} L_{tφ̃}^{l}(π\overline{ℂ})^{∗}1

= (π\overline{ℂ})^{∗}(ν_{t}(ψ) · lim_{l→∞} N_{t}^{l}1).

Hence, we obtain ψ_{0} = ν_{t}(ψ) · lim_{l→∞} N_{t}^{l}1.

Now, let τ be a Borel probability measure on J(G) such that N_{t}^{∗}τ = τ. Then for any ψ ∈ C(J(G)), we have τ(ψ) = ((N_{t}^{l})^{∗}τ)(ψ) = τ(N_{t}^{l}ψ) = ((exp(P (t)))^{l}) · τ(\frac{1}{(exp(P (t)))^{l}} N_{t}^{l}ψ) for any l ∈ ℕ. Since \frac{1}{(exp(P (t)))^{l}} N_{t}^{l}ψ → ν_{t}(ψ)·lim_{l→∞} N_{t}^{l}1 and N_{t}^{∗}τ = τ, we have τ(\frac{1}{(exp(P (t)))^{l}} N_{t}^{l}ψ) → ν_{t}(ψ) as l → ∞. Hence, it must be true that P (t) = 0, otherwise, we have τ(ψ) = 0 for all ψ or τ(ψ) is not bounded, both of which produce a contradiction. Hence, it follows that t = δ. Further, by the above argument, we obtain τ(ψ) = ν_{δ}(ψ) for any ψ ∈ C(J(G)).

Lemma 4.10. Let G = ⟨f_{1}, f_{2}, ⋯ f_{m}⟩ be a finitely generated expanding rational semigroup. Then, under the notation in Lemma 4.9, we have the following:

1. If there exists a t-conformal measure τ, then s(G) ≤ t ≤ δ.

2. If there exists a t-conformal measure τ satisfying the separating condition for {f_{1}, ⋯, f_{m}}, then t = δ and τ = ν.

Proof. First, we show the statement 1. By Lemma 4.8, we have N_{t}^{∗}τ ≥ τ. Hence, for each ψ ∈ C(J(G)) such that 0 ≤ ψ(z) for each z ∈ J(G), we have τ(N_{t}^{l}ψ) ≥ τ(ψ) for each l ∈ ℕ. Suppose that t > δ. Then, by Lemma 3.7-1, P (t) < 0. Hence, we obtain τ(N_{t}^{l}ψ) = (exp(P (t)))^{l}·τ(\frac{N_{t}^{l}τ}{(exp(P (t))^{l}}) → 0 as l → ∞, by Lemma 4.9-2. Hence, τ(ψ) = 0 for each ψ ∈ C(J(G)) such that 0 ≤ ψ(z) for each z ∈ J(G). This is a contradiction, since τ(1) = 1. Hence, t ≤ δ must hold. By Lemma 4.2, we have s(G) ≤ t. Hence, the statement 1 holds.

Next, we show the statement 2. By Lemma 4.8-2, we have N_{t}^{∗}τ = τ. Hence, by Lemma 4.9-3, it follows that t = δ and τ = ν.

Lemma 4.11. Let G be a rational semigroup and t ≥ 0 a number. Suppose that 0 < H^{t}(J(G)) < ∞. Let τ = \frac{H^{t}|_{J(G)}}{H^{t}(J(G))}. Then, τ is a t-conformal measure.

Proof. Suppose that t = 0. Then, each point z ∈ \overline{ℂ} satisfies H^{0}(\{z\}) = 1. Since we assume 0 < H^{t}(J(G)) < ∞, it follows that 1 ≤ ♯(J(G)) < ∞. Then, G consists of degree 1 maps and it is easy to see that τ is 0-conformal.

Suppose that t > 0. Then, H^{t} has no point mass. Let g ∈ G be an element. Step 1: For a critical point c of g in J(G), we have 0 = τ(g({c})) =

∫_{{c}} ∥g′∥^{t} dτ.

Step 2: Let W be a non-empty open set in \overline{ℂ} such that g : W → g(W) is a diffeomorphism. Let K be a compact subset of W and c > 0 a number. Let A be a Borel set such that A ⊂ {z ∈ g^{−1}(J(G)) | d(z, A) < c} ⊂ K ∩ g^{−1}(J(G)). Then, we show the following claim: Claim 1: we have H^{t}(g(A)) = ∫_{A} ∥g′∥^{t} dH^{t}.

To show this claim, let ϵ > 0 be a given number. Let K = ∑_{i=1}^{l} K_{i} be a disjoint union of Borel sets K_{i}, {z_{i}}_{i=1}^{l} a set with z_{i} ∈ K_{i} for each i, and ξ > 0 a real number, such that:

1. 1 − ϵ ≤ \frac{∥g′(z)∥}{∥g′(z_{i})∥} ≤ 1 + ϵ, for each z ∈ B(K_{i}, ξ), and

2. (1 − ϵ)∥g′(z_{i})∥ diam C ≤ diam g(C) ≤ (1 + ϵ)∥g′(z_{i})∥ diam C, for each subset C of B(K_{i}, ξ).

Let i ∈ ℕ (1 ≤ i ≤ l) be a fixed number. Let β be a number with 0 < β < ξ. Let {U_{p}}_{p=1}^{∞} be a sequence of sets such that A ∩ K_{i} ⊂ ∪_{p=1}^{∞}U_{p}, A ∩ K_{i} ∩ U_{p} ≠ ∅ for each p ∈ ℕ and diam U_{p} ≤ β for each p ∈ ℕ. Then, since β < ξ, we have U_{p} ⊂ B(K_{i}, ξ) for each p ∈ ℕ. Hence, we obtain g(A ∩ K_{i}) ⊂ ∪_{p=1}^{∞}g(U_{p}), diam g(U_{p}) ≤ (1 + ϵ)∥g′(z_{i})∥ diam U_{p} for each p ∈ ℕ and ∑_{p=1}^{∞}( diam g(U_{p}))^{t} ≤ (1 + ϵ)^{t}∥g′(z_{i})∥^{t} ∑_{p=1}^{∞}( diam U_{p})^{t}. This implies that H_{(1+ϵ)∥g′(z_{i})∥β}^{t}(g(A ∩ K_{i})) ≤ (1 + ϵ)^{t}∥g′(z_{i})∥^{t} ∑_{p=1}^{∞}( diam U_{p})^{t}. Hence, we obtain H_{(1+ϵ)∥g′(z_{i})∥β}^{t}(g(A∩K_{i})) ≤ (1+ϵ)^{t}∥g′(z_{i})∥^{t}H_{β}^{t}(A∩K_{i}). Then, we obtain H^{t}(g(A∩K_{i})) ≤ (1+ϵ)^{t}∥g′(z_{i})∥^{t}·H^{t}(A∩K_{i}), letting β → 0. Similarly, we obtain H^{t}(A ∩ K_{i}) ≤ (1 − ϵ)^{−t}∥g′(z_{i})∥^{−t}· H^{t}(g(A ∩ K_{i})). Hence, it follows that (1 − ϵ)^{t}∥g′(z_{i})∥^{t}H^{t}(A∩K_{i}) ≤ H^{t}(g(A∩K_{i})) ≤ (1+ϵ)^{t}∥g′(z_{i})∥^{t}H^{t}(A∩K_{i}). Moreover, (1 − ϵ)^{t}∥g′(z_{i})∥^{t}· H^{t}(A ∩ K_{i}) ≤ ∫_{A∩K_{i}} ∥g′∥^{t} dH^{t} ≤ (1 + ϵ)^{t}∥g′(z_{i})∥^{t}H^{t}(A ∩ K_{i}). Hence, we obtain

|H^{t}(g(A ∩ K_{i})) − ∫_{A∩K_{i}} ∥g′∥^{t} dH^{t}| ≤ ((1 + ϵ)^{t} − (1 − ϵ)^{t})∥g′(z_{i})∥^{t}H^{t}(A ∩ K_{i}).

This implies that |H^{t}(g(A))−∫_{A} ∥g′∥^{t} dH^{t}| ≤ ((1+ϵ)^{t} −(1−ϵ)^{t})·max_{z∈K} ∥g′(z)∥^{t} ·

∑_{i=1}^{l} H^{t}(A ∩ K_{i}). Since this inequality holds for each ϵ > 0, it follows that H^{t}(g(A)) = ∫_{A} ∥g′∥^{t} dH^{t}. Hence, we have shown Claim 1.

Step 3: Let B be a general Borel subset of g^{−1}(J(G)) such that g : B → g(B) is injective. Let B = ∑_{u=1}^{q}{c_{u}}⨿∑_{v=1}^{∞} B_{v} be a disjoint union of Borel sets such that each c_{u} is a critical point of g (if one exists) and for each B_{v} there exists an open set W_{v} in \overline{ℂ} such that \overline{B_{v}} ⊂ W_{v} and g : W_{v} → g(W_{v}) is a diffeomorphism. Then, by Steps 1 and 2, we obtain 0 = τ({g(c_{u})}) = ∫_{{c_{u}}} ∥g′∥^{t} dτ for each u, and τ(g(B_{v})) = ∫_{B_{v}} ∥g′∥^{t} dτ for each v. Combining this result with the fact that g : B → g(B) is injective, it follows that τ(g(B)) = ∑_{u=1}^{q} τ({g(c_{u})})+ ∑_{v=1}^{∞} τ(g(B_{v})) = ∑_{v=1}^{∞} ∫_{B_{v}} ∥g′∥^{t} dτ = ∫_{B} ∥g′∥^{t} dτ.

Hence, we have shown Lemma 4.11.

Lemma 4.12. Let G be a rational semigroup. Let τ be a t-subconformal measure for some t ∈ ℝ. Suppose that supp τ = J(G). Let g ∈ G. Then, each Borel subset A of g^{−1}(J(G)) with τ(A) = 0 has no interior points with respect to the induced topology on g^{−1}(J(G)).

Proof. Suppose there exists an open set U of \overline{ℂ} such that A ⊃ U ∩ g^{−1}(J(G)) ≠ ∅. Then, it follows that τ(g(U)) = τ(g(U) ∩ J(G)) = τ(g(U ∩ g^{−1}(J(G)))) ≤ ∫_{U∩g^{−1}(J(G))} ∥g′∥^{t} dτ = 0. This is a contradiction because we assume supp τ = J(G).

The following proposition is needed to show Main Theorem B.

Proposition 4.13. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated expanding rational semigroup. Let f̃: Σ_{m} × \overline{ℂ} → Σ_{m} × \overline{ℂ} be the skew product map associated with \{f_{1}, ⋯, f_{m}\}. Let δ be a number in Lemma 3.7. Let ν := (π\overline{ℂ})_{∗}(ν̃_{δφ̃}).

Suppose that 0 < H^{δ}(J(G)). Then, we have H^{δ}(J(G)) < ∞, ν = \frac{H^{δ}|_{J(G)}}{H^{δ}(J(G))}, and ν is a δ-conformal measure satisfying the separating condition with respect to {f_{1}, ⋯, f_{m}}. Furthermore, f_{i}^{−1}(J(G)) ∩ f_{j}^{−1}(J(G)) is nowhere dense in f_{j}^{−1}(J(G)), for each (i, j) with i ≠ j.

Proof. By Lemma 3.11 and Lemma 3.16, we obtain that ν is a δ-subconformal measure, H^{δ}|_{J(G)} is absolutely continuous with respect to ν, and H^{t}(J(G)) <

∞. Let τ := \frac{H^{δ}|_{J(G)}}{H^{δ}(J(G))}. Let φ ∈ L^{1}(ν) be the density function such that τ(A) = ∫_{A} φ dν for any Borel subset A of J(G). We show the following claim: Claim 1: We have (φ ∘ π\overline{ℂ} ∘ f̃)(z) ≥ (φ ∘ π\overline{ℂ})(z) for almost every z ∈ J̃(f̃) with respect to ν̃ := ν̃_{δφ}.

To show this claim, let j (1 ≤ j ≤ m) be a number and A an open subset of J(G) such that we can take a well-defined inverse branch γ of f_{j}^{−1} on A. By Lemma 4.11, τ is δ-conformal. Hence, for each Borel subset B of A, we have τ(B) = ∫_{γ(B)} ∥f′_{j}∥^{δ} dτ = ∫_{γ(B)} ∥f′_{j}∥^{δ}φ dν. Moreover, by Lemma 3.11, we have ν is δ-subconformal. Hence, we obtain τ(B) = ∫_{B} φ dν = ∫_{A}(φ ∘ f_{j} ∘ γ) · (1_{γ(B)} ∘ γ) dν = ∫_{A}(φ ∘ f_{j}) · 1_{γ(B)} d(γ_{∗}(ν|_{A})) ≤ ∫_{γ(B)} ∥f′_{j}∥^{δ}((φ ∘ f_{j})) dν. Hence, we obtain φ(x) ≤ (φ ∘ f_{j})(x) for almost every x ∈ γ(A) with respect to ν. It follows that for each j = 1, ⋯, m, we have φ(x) ≤ (φ ∘ f_{j})(x) for almost every x ∈ f_{j}^{−1}(J(G)) with respect to ν. This implies that for each j = 1, ⋯, m, we have (φ∘π\overline{ℂ})(z) ≤ (φ∘f_{j}∘π\overline{ℂ})(z) for almost every z ∈ π−1 ; \overline{ℂ}f_{j}^{−1}(J(G)) with respect to ν̃. Since J̃(f̃) = ∪_{j=1}^{m}Σ_{m}(j) ∩ J̃(f̃) and Σ_{m}(j) ∩ J̃(f̃) ⊂ π−1 ; \overline{ℂ}(f_{j}^{−1}(J(G))) (the latter follows from π\overline{ℂ}f̃((w, x)) = f_{j}(x) = f_{j}(π\overline{ℂ}((w, x))) for each (w, x) ∈ Σ_{m}(j) ∩ J̃(f̃)), it follows that (φ ∘ π\overline{ℂ}(z)) ≤ (φ ∘ π\overline{ℂ} ∘ f̃)(z) for almost every z ∈ J̃(f̃) with respect to ν̃. Hence, we have shown Claim 1.

By Claim 1, we have (φ ∘ π\overline{ℂ}(z)) ≤ (φ ∘ π\overline{ℂ} ∘ f̃)(z) for almost every z ∈ J̃(f̃) with respect to αν̃, where α is the function in Lemma 3.7. Let ψ = φ ∘ π\overline{ℂ}. Then, we obtain for each n ∈ ℕ, ψ(z) ≤ \frac{1}{n} ∑_{j=0}^{n−1} ψ ∘ f̃^{j}(z) for almost every z with respect to αν̃. Note that by Lemma 3.6-3, the measure αν̃ is f̃-invariant. Hence, by Birkhoff’s ergodic theorem (see [DGS]), we have ψ(z) ≤ lim_{n→∞} \frac{1}{n} ∑_{j=0}^{n−1}(ψ ∘ f̃^{j})(z) for almost every z with respect to αν̃. Since ∫ ψα dν̃ = ∫ lim_{n→∞} \frac{1}{n} ∑_{j=0}^{n−1}(ψ ∘ f̃^{j})(z) αdν̃(z), which follows from Birkhoff’s ergodic theorem again, it follows that ψ(z) = lim_{n→∞} \frac{1}{n} ∑_{j=0}^{n−1}(ψ ∘ f̃^{j})(z) for almost every z with respect to αν̃. Since αν̃ is ergodic (Lemma 3.6-3), then there exists a constant c such that lim_{n→∞} \frac{1}{n} ∑_{j=0}^{n−1}(ψ ∘ f̃^{j})(z) = c for almost every z with respect to αν̃. Hence, it follows that ψ(z) = c for almost every z with respect to ν̃. Since τ and ν are probability measures, it follows that c = 1. Hence, τ = ν. Since N_{δ}^{∗}ν = ν (Lemma 4.9-1) and τ is δ-conformal (Lemma 4.11), by Lemma 4.8-3 it follows that ν = τ is a δ-conformal measure satisfying the separating condition with respect to {f_{1}, ⋯, f_{m}}. Since supp ν = J(G) (Main Theorem A), by Lemma 4.12, it follows that f_{i}^{−1}(J(G)) ∩ f_{j}^{−1}(J(G)) is nowhere dense in f_{j}^{−1}(J(G)) for each (i, j) with i ≠ j.

Hence, we have shown Proposition 4.13.

Example 4.14. Let f_{1}(z) = z^{2}, f_{2}(z) = \frac{z^{2}}{4} and f_{3}(z) = \frac{z^{2}}{3} . Let G = ⟨f_{1}, f_{2}, f_{3}⟩ and f̃: Σ_{3} × \overline{ℂ} → Σ_{3} × \overline{ℂ} be the skew product map with respect to \{f_{1}, f_{2}, f_{3}\}. Then, it is easy to see J(⟨f_{1}, f_{2}⟩) = {z | 1 ≤ |z| ≤ 4}. Since f_{3}^{−1}(J(⟨f_{1}, f_{2}⟩)) = \{z | \sqrt{3} ≤ |z| ≤ 2\sqrt{3}\} ⊂ J(⟨f_{1}, f_{2}⟩), we have J(G) = \{z | 1 ≤ |z| ≤ 4.\}. Then, P (G) = {0, ∞} ⊂ F (G). By Theorem 2.6 in [S2], we find that G is expanding. Furthermore, we have 0 < H^{2}(J(G)) < ∞ and H^{2}(f_{1}^{−1}(J(G))∩f_{3}^{−1}(J(G))) > 0. Hence, by Proposition 4.13, the number δ in Lemma 3.7 for f̃ satisfies δ > 2.

5 Main Theorem B

In this section, we demonstrate Main Theorem B. First, we need the following notation.

Definition 5.1. Let G = ⟨f_{1}, ⋯, f_{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 system {f_{1}, ⋯, f_{m}} if f_{j}^{−1}(U) ⊂ U for each j = 1, ⋯, m and {f_{j}^{−1}(U)}_{j=1}^{m} are mutually disjoint.

Lemma 5.2. 1. If a rational semigroup G = ⟨f_{1}, ⋯, f_{m}⟩ satisfies the open set condition with U and ♯J(G) ≥ 3, then J(G) ⊂ \overline{U}.

2. If a rational semigroup G = ⟨f_{1}, ⋯, f_{m}⟩ is expanding and if G satisfies the open set condition with U, then J(G) ⊂ \overline{U}.

Proof. By Lemma 2.3 (f) in [S5] and Lemma 3.2, it is easy to see the statement.

To show Main Theorem B, we need the following key lemma.

Lemma 5.3. Let G = ⟨f_{1}, ⋯, f_{m}⟩ be a finitely generated expanding rational semigroup satisfying the open set condition with an open set U with respect to {f_{1}, ⋯, f_{m}}. Then, we have the following.

1. There exists a positive constant C such that for each r with 0 < r < diam \overline{ℂ} and each x ∈ J(G), we have C^{−1}r^{δ} ≤ ν(B(x, r)) ≤ Cr^{δ}. Furthermore, 0 < H^{δ}(J(G)) < ∞ and dim_{H}(J(G)) = \overline{dim}_{B}(J(G)) = δ.

2. Suppose that there exists a t-conformal measure τ. Then, there exists a positive constant C_{0} such that for any r with 0 < r < diam \overline{ℂ} and any x ∈ J(G), we have C_{0}^{−1}r^{t} ≤ τ(B(x, r)) ≤ C_{0}r^{t}. Furthermore, we have 0 < H^{t}(J(G)) < ∞ and dim_{H}(J(G)) = t = δ. Moreover, ν and τ are absolutely continuous with respect to each other.

To show this lemma, we need several other lemmas(Lemma 5.4-Lemma 5.15). We suppose the assumption of Lemma 5.3, until the end of the proof of Lemma 5.3.

Preparation to show Lemma 5.3:

1. First, we may assume that \overline{U} ∩P (G) = ∅. For, let V be a ϵ_{0}-neighborhood of P (G) with respect to the hyperbolic metric on F (G). Then, for each g ∈ G, we have g(V) ⊂ V, which implies that W := U \ \overline{V} satisfies f_{j}^{−1}(W) ⊂ W, for each j = 1, ⋯, m, and {f_{j}^{−1}(W)}_{j} are mutually disjoint. Hence we may assume the above.

Assuming that \overline{U} ∩ P (G) = ∅, take a number ϵ > 0 such that B(\overline{U}, 2ϵ) ∩ P (G) = ∅. Then for each y ∈ \overline{U} and any g ∈ G, we can take well-defined inverse branches of g^{−1} on B(y, 2ϵ).

2. Let \overline{U} = ∑_{j=1}^{k} K_{j} be a measurable partition such that for each j = 1, ⋯, k, we have int K_{j} ≠ ∅ and diam K_{j} ≤ \frac{1}{10}ϵ. We take a point z_{j} ∈ K_{j}, for each j = 1, ⋯, k.

3. To show Lemma 5.3, we may assume that: for each j = 1, ⋯, k and each w ∈ W^{∗}, if γ is an inverse branch of f_{w}^{−1} on B(z_{j}, 2ϵ), then we have

diamγ(A) ≤ (\frac{1}{10})^{|w|} · diamA,  (3)

for each subset A of B(z_{j}, 2ϵ). For, for each n ∈ ℕ, G_{n} (see the notation in section 2.1) satisfies J(G_{n}) = J(G). Further, if we use ν^{n} to denote the Borel probability measure on J(G_{n}) = J(G) constructed by the generator system {f_{w} | |w| = n} of G_{n}, for which the construction method is the same as that for ν from {f_{j}}, then ν^{n} satisfies (N_{δ_{n}}^{n})^{∗}ν^{n} = ν^{n} for some δ_{n} ∈ ℝ. Since ν satisfies (N_{δ}^{n})^{∗}ν = ν, by Lemma 4.9 we obtain δ_{n} = δ and ν^{n} = ν. Moreover, since G is expanding, by the Koebe distortion theorem there exist numbers ϵ′ > 0 and n ∈ ℕ such that if γ is a well-defined inverse branch of f_{w}^{−1} on B(z, 2ϵ′), where |w| = n and z ∈ J(G), then for any subset A of B(z, 2ϵ′), diam γ(A) ≤ \frac{1}{10} diam A. Let U′ := U ∩ B(J(G), ϵ′). Then, for each w ∈ {1, ⋯, m}^{n}, f_{w}^{−1}(U′) ⊂ U′ and {f_{w}^{−1}(U′)}_{w:|w|=n} are mutually disjoint. Hence, we may assume the above.

4. Let r > 0 be fixed. There exists a number s ∈ ℕ with s ≥ 3 such that for each j = 1, ⋯, k and each w ∈ W^{∗} with |w| ≥ s−1, we have diam γ(K_{j}) ≤ r, for each well-defined inverse branch γ of f_{w}^{−1} on B(z_{j}, 2ϵ). We fix such an s. Let A be the set of all (γ, K_{j}) that satisfies j ∈ {1, ⋯, k}, γ is a well-defined inverse branch of f_{w}^{−1} on B(z_{j}, 2ϵ) for some w ∈ {1, ⋯, m}^{s}, and γ(K_{j}) ∩ B(x, r) ≠ ∅.

Then, we have the following:

Lemma 5.4. B(x, r) ∩ J(G) = B(x, r) ∩ ⋃_{(γ,K_{j})∈A} γ(J(G) ∩ K_{j}).

Proof. Since J(G) = ∪_{j=1}^{m}f_{j}^{−1}(J(G)) (Lemma 2.4 in [S5]) and J(G) ⊂ \overline{U} (Lemma 5.2), it is easy to see the statement.

Definition 5.5. 1. Let (γ, K_{j}) ∈ A be any element such that γ is an inverse branch of f_{w}^{−1}, where w = (w_{1}, ⋯, w_{s}) ∈ {1, ⋯, m}^{s}. Then, we decompose γ as γ = γ_{1} ⋯ γ_{s}, where, for each i = 1, ⋯, s, we use γ_{i} to denote the inverse branch of f_{w_{i}}^{−1} on B(γ_{i+1} ⋯ γ_{s}(z_{j}), 2ϵ).

2. For each A = (γ, K_{j}) ∈ A, let l(A) be the minimum of l ∈ ℕ that satisfies 3 ≤ l ≤ s and if γ_{l} ⋯ γ_{s}(K_{j})∩K_{i} ≠ ∅, then diam γ_{1} ⋯ γ_{l−1}(K_{i}) ≤ r. Note that by (3), we have γ_{1} ⋯ γ_{l−1} is defined on K_{i} with γ_{l} ⋯ γ_{s}(K_{j})∩K_{i} ≠ ∅. Moreover, note that according to the choice of s, l(A) exists, for each A ∈ A.

Lemma 5.6. Let A = (γ, K_{j}) ∈ A. If

r < min{diamγ′_{1}(K_{i}) | (γ′, K_{t}) ∈ A, i ∈ {1, ⋯, k}, K_{i} ⊂ B(γ′_{2} ⋯ γ′_{s}(z_{t}), 2ϵ)}, (4)

then there exists an element K_{i} such that γ_{l(A)−1} ⋯ γ_{s}(K_{j}) ∩ K_{i} ≠ ∅ and diam γ_{1} ⋯ γ_{l(A)−2}(K_{i}) > r.

Proof. If l(A) ≥ 4, then it is trivial. If l(A) = 3, then by (4), the above is true.

Remark 6. For the rest, we assume (4). To show Lemma 5.3, we may make this assumption.

Definition 5.7. For any A = (γ, K_{j}) ∈ A, we set

Γ_{A} := {(γ_{1} ⋯ γ_{l(A)−1}, K_{i}) | γ_{l(A)} ⋯ γ_{s}(K_{j}) ∩ K_{i} ≠ ∅}.

Further, we set Γ = ∪_{A∈A}Γ_{A} (disjoint union).

Let B_{1} and B_{2} be two elements in Γ with B_{1} = (γ_{1} ⋯ γ_{l(A)−1}, K_{i_{1}}) ∈ Γ_{A} and B_{2} = (γ′_{1} ⋯ γ′_{l(A′)−1}, K_{i_{2}}) ∈ Γ_{A′}, where A = (γ, K_{j(A)}) ∈ A and A′ = (γ′, K_{j(A′)}) ∈ A. Then,

1. We write B_{1} ∼ B_{2} if and only if K_{i_{1}} = K_{i_{2}} and γ_{1} ⋯ γ_{l(A)−1} = γ′_{1} ⋯ γ′_{l(A′)−1} on B(γ_{l(A)} ⋯ γ_{s}(z_{j(A)}), 2ϵ) ∩ B(γ′_{l(A′)} ⋯ γ′_{s}(z_{j(A′)}), 2ϵ). Note that this ∼ is an equivalence relation on Γ, by (3) and the uniqueness theorem.

2. We write B_{1} ≼ B_{2} if and only if

γ_{1} ⋯ γ_{l(A)−1}(intK_{i_{1}}) ∩ γ′_{1} ⋯ γ′_{l(A′)−1}(intK_{i_{2}}) ≠ ∅

and l(A) ≤ l(A′).

For any two elements B and B′ in Γ, we write B ≼≼ B′ if and only if there exists a sequence {B_{l}}_{l=1}^{v} in Γ such that B = B_{1} ≼ ⋯ ≼ B_{v} = B′.

Lemma 5.8. Let B_{1} and B_{2} be two elements in Γ with B_{1} = (γ_{1} ⋯ γ_{l(A)−1}, K_{i_{1}}) ∈ Γ_{A} and B_{2} = (γ′_{1} ⋯ γ′_{l(A′)−1}, K_{i_{2}}) ∈ Γ_{A′}, where A = (γ, K_{j(A)}) ∈ A and A′ = (γ′, K_{j(A′)}) ∈ A. Suppose that B_{1} ≼ B_{2}. Then, we have the following.

1. If l(A) = l(A′), then B_{1} ∼ B_{2}.

2. If l(A) < l(A′), then

(a) int K_{i_{1}} ∩ γ′_{l(A)} ⋯ γ′_{l(A′)−1}( int K_{i_{2}}) ≠ ∅ and

(b) γ_{1} ⋯ γ_{l(A)−1} = γ′_{1} ⋯ γ′_{l(A)−1} on B(γ_{l(A)} ⋯ γ_{s}(z_{j(A)}), 2ϵ) ∩ B(γ′_{l(A)} ⋯ γ′_{s}(z_{j(A′)}), 2ϵ).

Proof. First, we show 2. Under the assumption of 2, suppose that γ_{1} ⋯ γ_{l(A)−1} is an inverse branch of f_{w_{1}}^{−1} ⋯ f_{w_{l(A)−1}}^{−1} and that γ′_{1} ⋯ γ′_{l(A′)−1} is an inverse branch of f_{w′_{1}}^{−1} ⋯ f_{w′_{l(A′)−1}}^{−1}. By the open set condition, it follows that w_{j} = w′_{j},

for each j = 1, ⋯, l(A) − 1. Hence, 2a holds.

Next, take a point z ∈ γ_{1} ⋯ γ_{l(A)−1}( int K_{i_{1}}) ∩ γ′_{1} ⋯ γ′_{l(A′)−1}( int K_{i_{2}}). Let a := f_{w_{l(A)−1}} ⋯ f_{w_{1}}(z). Then, we have a ∈ int K_{i_{1}} ∩ γ′_{l(A)} ⋯ γ′_{l(A′)−1}( int K_{i_{2}}).

Furthermore, each of γ_{1} ⋯ γ_{l(A)−1} and γ′_{1} ⋯ γ′_{l(A)−1} is a well-defined inverse branch of (f_{w_{l(A)−1}} ⋯ f_{w_{1}})^{−1} on B(a, ϵ) and maps a to z. Hence, they are equal on B(a, ϵ). By the uniqueness theorem, we obtain 2b.

We can show 1 using the same method as above.

Lemma 5.9. If B and B′ are two elements of Γ such that B ≼≼ B′ and B′ ≼≼ B, then B ∼ B′.

Proof. There exists a sequence {B_{j}}_{j=1}^{v} in Γ such that B = B_{1} ≼ ⋯ ≼ B_{u} = B′ ≼ ⋯ ≼ B_{v} = B. Suppose B_{j} ∈ Γ_{A_{j}}, for each j = 1, ⋯, v. Then we have l(A_{1}) ≤ ⋯ ≤ l(A_{v}) = l(A_{1}). By Lemma 5.8, we obtain B_{j} ∼ B_{j+1}, for each j = 1, ⋯, u − 1.

Lemma 5.10. If B_{1} ∼ B_{2}, B_{3} ∼ B_{4} and B_{1} ≼ B_{3}, then B_{2} ≼ B_{4}.

Proof. This is easy to see, from the definitions of “∼” and “≼”, by using (3).

Definition 5.11. For any B ∈ Γ, we use [B] ∈ Γ/∼ to denote the equivalence class of B, with respect to the equivalence relation ∼ in Γ.

Let [B_{1}] and [B_{2}] be two elements of Γ/ ∼, where B_{1}, B_{2} ∈ Γ. We write [B_{1}] ≼ [B_{2}] if and only if B_{1} ≼ B_{2}. Note that this is well defined by Lemma 5.10. Furthermore, we write [B_{1}] ≤ [B_{2}] if and only if B_{1} ≼≼ B_{2}. Note that this is also well defined by Lemma 5.10 and that the “≤” determines a partial order in Γ/∼, by Lemma 5.9.

Lemma 5.12. Let q ∈ ℕ be an integer with q ≥ 2. Let {B_{j}}_{j=1}^{q} be a sequence in Γ such that B_{1} ≼ ⋯ ≼ B_{q} and B_{j} ≁ B_{j+1}, for each j = 1, ⋯, q − 1. Suppose that for each j = 1, ⋯, q, we have B_{j} ∈ Γ_{A_{j}}, A_{j} = (γ^{j}, K_{t_{j}}) ∈ A and B_{j} = (γ_{1}^{j} ⋯ γ_{l(A_{j})−1}^{j}, K_{i_{j}}). Then, we have the following.

1. γ_{l(A_{1})}^{q} ⋯ γ_{l(A_{q})−1}^{q}(K_{i_{q}}) ⊂ B(K_{i_{1}}, ∑_{j=1}^{q−1}(\frac{1}{10})^{j} \frac{1}{10}ϵ).

2. γ_{1}^{1} ⋯ γ_{l(A_{1})−1}^{1} = γ_{1}^{q} ⋯ γ_{l(A_{1})−1}^{q} on V := B(γ_{l(A_{1})}^{1} ⋯ γ_{s}^{1}(z_{t_{1}}), 2ϵ) ∩ B(γ_{l(A_{1})}^{q} ⋯ γ_{s}^{q}(z_{t_{q}}), 2ϵ). (Note that by 1, we have V ≠ ∅.)

Proof. We will show the statement by induction on q. If q = 2, then the statement follows from Lemma 5.8 and (3). Let q ≥ 3. Suppose that the statement holds for each q′ with 2 ≤ q′ ≤ q − 1. By Lemma 5.8, we have l(A_{j}) < l(A_{j+1}), for each j = 1, ⋯, q − 1. By the hypothesis of induction, we have the following claim. Claim 1:

1. γ_{l(A_{2})}^{q} ⋯ γ_{l(A_{q})−1}^{q}(K_{i_{q}}) ⊂ B(K_{i_{2}}, ∑_{j=1}^{q−2}(\frac{1}{10})^{j} \frac{1}{10}ϵ).

2. γ_{1}^{2} ⋯ γ_{l(A_{2})−1}^{2} = γ_{1}^{q} ⋯ γ_{l(A_{2})−1}^{q} on B(γ_{l(A_{2})}^{2} ⋯ γ_{s}^{2}(z_{t_{2}}), 2ϵ) ∩ B(γ_{l(A_{2})}^{q} ⋯ γ_{s}^{q}(z_{t_{q}}), 2ϵ).

Combining Claim 1 with (3), we obtain

γ_{l(A_{1})}^{q} ⋯ γ_{l(A_{q})−1}^{q}(K_{i_{q}}) ⊂ B(γ_{l(A_{1})}^{2} ⋯ γ_{l(A_{2})−1}^{2}(K_{i_{2}}), ∑_{j=2}^{q−1}(\frac{1}{10})^{j} \frac{1}{10}ϵ). (5)

Moreover, by Lemma 5.8 and (3), we have γ_{l(A_{1})}^{2} ⋯ γ_{l(A_{2})−1}^{2}(K_{i_{2}}) ⊂ B(K_{i_{1}}, \frac{1}{10} \frac{1}{10}ϵ). Hence, we obtain

γ_{l(A_{1})}^{q} ⋯ γ_{l(A_{q})−1}^{q}(K_{i_{q}}) ⊂ B(K_{i_{1}}, ∑_{j=1}^{q−1}(\frac{1}{10})^{j} \frac{1}{10}ϵ).  (6)

Hence, the statement 1 in our lemma holds for q.

Next, we will show that the statement 2 in our lemma holds for q. Let us consider 2 in Claim 1. By the open set condition, for each j = 1, ⋯, l(A_{1}) − 1, there exists a number α_{j} ∈ {1, ⋯, m} such that each of γ_{j}^{2} and γ_{j}^{q} is an inverse branch of f_{α_{j}}^{−1}. Hence, we obtain

γ_{l(A_{1})}^{2} ⋯ γ_{l(A_{2})−1}^{2} = γ_{l(A_{1})}^{q} ⋯ γ_{l(A_{2})−1}^{q}  (7)

on V_{0} := B(γ_{l(A_{2})}^{2} ⋯ γ_{s}^{2}(z_{t_{2}}), 2ϵ) ∩ B(γ_{l(A_{2})}^{q} ⋯ γ_{s}^{q}(z_{t_{q}}), 2ϵ).

Let β := γ_{l(A_{1})}^{2} ⋯ γ_{l(A_{2})−1}^{2} = γ_{l(A_{1})}^{q} ⋯ γ_{l(A_{2})−1}^{q} on V_{0}. Then by 2 in Claim 1, we obtain γ_{1}^{2} ⋯ γ_{l(A_{1})−1}^{2} = γ_{1}^{q} ⋯ γ_{l(A_{1})−1}^{q} on β(V_{0}). Hence, by the uniqueness theorem, we get

γ_{1}^{2} ⋯ γ_{l(A_{1})−1}^{2} = γ_{1}^{q} ⋯ γ_{l(A_{1})−1}^{q}  (8)

on B(γ_{l(A_{1})}^{2} ⋯ γ_{s}^{2}(z_{t_{2}}), 2ϵ) ∩ B(γ_{l(A_{1})}^{q} ⋯ γ_{s}^{q}(z_{t_{q}}), 2ϵ).

Moreover, by Lemma 5.8, we have the following claim. Claim 2:

1. int K_{i_{1}} ∩ γ_{l(A_{1})}^{2} ⋯ γ_{l(A_{2})−1}^{2}( int K_{i_{2}}) ≠ ∅.

2. γ_{1}^{1} ⋯ γ_{l(A_{1})−1}^{1} = γ_{1}^{2} ⋯ γ_{l(A_{1})−1}^{2} on B(γ_{l(A_{1})}^{1} ⋯ γ_{s}^{1}(z_{t_{1}}), 2ϵ) ∩ B(γ_{l(A_{1})}^{2} ⋯ γ_{s}^{2}(z_{t_{2}}), 2ϵ).

Combining 1 in Claim 2 with (3), we obtain

d(γ_{l(A_{1})}^{1} ⋯ γ_{s}^{1}(z_{t_{1}}), γ_{l(A_{1})}^{2} ⋯ γ_{s}^{2}(z_{t_{2}})) ≤ \frac{1}{5}ϵ.  (9)

Furthermore, by (6) and (3), we obtain

d(γ_{l(A_{1})}^{q} ⋯ γ_{s}^{q}(z_{t_{q}}), γ_{l(A_{1})}^{1} ⋯ γ_{s}^{1}(z_{t_{1}})) ≤ \frac{1}{10}ϵ + \frac{1}{10}ϵ + ∑_{j=1}^{q−1}(\frac{1}{10})^{j} \frac{1}{10}ϵ ≤ \frac{3}{10}ϵ.(10)

Hence, by (9) and (10), we get W := ⋂_{j=1,2,q} B(γ_{l(A_{1})}^{j} ⋯ γ_{s}^{j}(z_{t_{j}}), 2ϵ) ≠ ∅. By 2 in Claim 2 and (8), then on W, γ_{1}^{1} ⋯ γ_{l(A_{1})−1}^{1} = γ_{1}^{q} ⋯ γ_{l(A_{1})−1}^{q}. Hence, by the uniqueness theorem, it follows that γ_{1}^{1} ⋯ γ_{l(A_{1})−1}^{1} = γ_{1}^{q} ⋯ γ_{l(A_{1})−1}^{q} on B(γ_{l(A_{1})}^{1} ⋯ γ_{s}^{1}(z_{t_{1}}), 2ϵ) ∩ B(γ_{l(A_{1})}^{q} ⋯ γ_{s}^{q}(z_{t_{q}}), 2ϵ). Hence, the statement 2 in our lemma holds for q. Hence, the induction is completed.

Lemma 5.13. Using the same assumption as for Lemma 5.12, it holds that γ_{l(A_{1})}^{q} ⋯ γ_{s}^{q}(K_{t_{q}}) ⊂ B(K_{i_{1}}, \frac{1}{5}ϵ).

Proof. By Lemma 5.12 and (3), we obtain

γ_{l(A_{1})}^{q} ⋯ γ_{s}^{q}(K_{t_{q}}) ⊂ B(K_{i_{1}}, (∑_{j=1}^{∞}(\frac{1}{10})^{j})\frac{1}{10}ϵ + \frac{1}{10}ϵ) ⊂ B(K_{i_{1}}, \frac{1}{5}ϵ).

Definition 5.14. Let {[m_{1}], ⋯, [m_{p}]} be the set of all minimal elements of (Γ/ ∼, ≤), where, for each i = 1, ⋯, p, m_{i} ∈ Γ_{R_{i}}, R_{i} = (γ^{i}, K_{u_{i}}) ∈ A and m_{i} = (γ_{1}^{i} ⋯ γ_{l(R_{i})−1}^{i}, K_{v_{i}}). Furthermore, for any i = 1, ⋯, p, we use η^{i} : π−1 ; \overline{ℂ}(B(γ_{l(R_{i})}^{i} ⋯ γ_{s}^{i}(z_{u_{i}}), 2ϵ)) → Σ_{m} × \overline{ℂ} to denote the inverse branch of (f̃^{l(R_{i})−1})^{−1} such that η^{i}((w, y)) = (w^{i}w, γ_{1}^{i} ⋯ γ_{l(R_{i})−1}^{i}(y)) for each (w, y) ∈

π−1 ; \overline{ℂ} (B(γ_{l(R_{i})}^{i} ⋯ γ_{s}^{i}(z_{u_{i}}), 2ϵ)), where w^{i} ∈ W^{∗} is a word satisfying |w^{i}| =

l(R_{i}) − 1 and γ_{1}^{i} ⋯ γ_{l(R_{i})−1}^{i} is an inverse branch of f_{w^{i}}^{−1}.

Lemma 5.15. 1. π−1 ; \overline{ℂ}(B(x, r)) ∩ J̃(f̃) ⊂ ⋃_{i=1}^{p} η^{i}(π−1 ; \overline{ℂ}(B(K_{v_{i}}, \frac{1}{5}ϵ)) ∩ J̃(f̃)).

2. B(x, r) ∩ J(G) ⊂ ⋃_{i=1}^{p} γ_{1}^{i} ⋯ γ_{l(R_{i})−1}^{i}(B(K_{v_{i}}, \frac{1}{5}ϵ) ∩ J(G)).

Proof. Let (w, z) ∈ π−1 ; \overline{ℂ}(B(x, r)) ∩ J̃(f̃) be a point. By Lemma 5.2, there exists a number j such that π\overline{ℂ}f̃^{s}((w, z)) ∈ K_{j}. Let η : π−1 ; \overline{ℂ}B(z_{j}, 2ϵ) → Σ_{m} × \overline{ℂ} be an inverse branch of (f̃^{s})^{−1} such that η((w′, x′)) = ((w|s) · w′, γ(x′)) where γ is an inverse branch of f_{w|s}^{−1}. Then, we have (w, z) ∈ η(π−1 ; \overline{ℂ}(B(z_{j}, 2ϵ))) and A := (γ, K_{j}) ∈ A. Let B = (γ_{1} ⋯ γ_{l(A)−1}, K_{i_{1}}) ∈ Γ_{A} be an element. Then, there exists a number i with 1 ≤ i ≤ p such that [m_{i}] ≤ [B]. We will show the following claim:

Claim 1: (w, z) ∈ η^{i}(π−1 ; \overline{ℂ}(B(K_{v_{i}}, \frac{1}{5}ϵ)) ∩ J̃(f̃)).

To show this claim, we consider the following two cases: Case 1: B ∼ m_{i} Case 2: There exists a sequence (B_{j})_{j=1}^{q} in Γ such that m_{i} = B_{1} ≼ B_{2} ≼ ⋯ ≼ B_{q} = B and B_{j} ≁ B_{j+1} for each j = 1, ⋯ q − 1.

Suppose that we have Case 2. Let y = π\overline{ℂ}(f̃^{s}((w, z))) ∈ K_{j} ∩ J(G). Then, we have z = γ(y) = γ_{1} ⋯ γ_{l(R_{i})−1} · γ_{l(R_{i})} ⋯ γ_{s}(y). By Lemma 5.13, we have γ_{l(R_{i})} ⋯ γ_{s}(y) ∈ B(K_{v_{i}}, \frac{1}{5}ϵ) ∩ J(G). Furthermore, by Lemma 5.12-2, we have γ_{1} ⋯ γ_{l(R_{i})−1} = γ_{1}^{i} ⋯ γ_{l(R_{i})−1}^{i} on B(K_{v_{i}}, \frac{1}{5}ϵ). Combining this with B(K_{v_{i}}, \frac{1}{5}ϵ)∩ U ≠ ∅ and the open set condition, we get w|(l(R_{i}) − 1) = w^{i}. By these argu-

ments, we obtain (w, z) = η^{i}(f̃^{l(R_{i})−1}((w, z))) ∈ η^{i}(π−1 ; \overline{ℂ}(B(K_{v_{i}}, \frac{1}{5}ϵ) ∩ J̃(f̃)).

Suppose that we have Case 1. Then, by the open set condition, the statement in Claim 1 is true. Hence, we have shown Claim 1.

By Claim 1, it follows that the statement of our lemma is true.

We now demonstrate Lemma 5.3. Proof of Lemma 5.3. Let A = (γ, K_{j}) ∈ A and B = (γ_{1} ⋯ γ_{l(A)−1}, K_{i}) ∈ Γ_{A}. By Lemma 5.6 and Remark 6, there exists a number u ∈ ℕ with 1 ≤ u ≤ k such that γ_{l(A)−1} ⋯ γ_{s}(K_{j}) ∩ K_{u} ≠ ∅ and diam γ_{1} ⋯ γ_{l(A)−2}(K_{u}) > r. Then, by the Koebe distortion theorem, there exists a positive constant C_{1} = C_{1}(min_{j} diamK_{j}, ϵ), which is independent of r, s and x ∈ J(G), such that ∥(γ_{1} ⋯ γ_{l(A)−2})′(z)∥ ≥ C_{1}r for each z ∈ B(γ_{l(A)−1} ⋯ γ_{s}(z_{j}), ϵ). Hence, there exists a positive constant C_{2} = C_{2}(C_{1}, G) such that

∥(γ_{1} ⋯ γ_{l(A)−1})′(z)∥ ≥ C_{2}r,

for each z ∈ B(γ_{l(A)} ⋯ γ_{s}(z_{j}), ϵ). Combining this with

K_{i} ⊂ B(γ_{l(A)} ⋯ γ_{s}(z_{j}), \frac{1}{5}ϵ),

which follows from (3), we obtain ∥(γ_{1} ⋯ γ_{l(A)−1})′(z)|| ≥ C_{2}r, for each z ∈ K_{i}. Hence, it follows that there exists a positive constant C_{3}, which is independent of r, s and x ∈ J(G), such that

meas_{2}(γ_{1} ⋯ γ_{l(A)−1}(intK_{i})) ≥ C_{3}r^{2},  (11)

where meas_{2} denotes the 2-dimensional Lebesgue measure. We now show the following claim: Claim: γ_{1}^{i} ⋯ γ_{l(R_{i})−1}^{i}(intK_{v_{i}}) ⊂ B(x, 3r), for each i = 1, ⋯, p.

To show this claim, since γ_{1}^{i} ⋯ γ_{s}^{i}(K_{u_{i}})∩B(x, r) ≠ ∅ and γ_{l(R_{i})}^{i} ⋯ γ_{s}^{i}(K_{u_{i}})∩ K_{v_{i}} ≠ ∅, we obtain γ_{1}^{i} ⋯ γ_{l(R_{i})−1}^{i}(K_{v_{i}}) ∩ B(x, 2r) ≠ ∅. Combining this with the fact that diam (γ_{1}^{i} ⋯ γ_{l(R_{i})−1}^{i}(K_{v_{i}})) ≤ r, it follows that the above claim holds.

Since {[m_{1}], ⋯, [m_{p}]} is the set of minimal elements of (Γ/∼, ≤), we find that {γ_{1}^{i} ⋯ γ_{l(R_{i})−1}^{i}(intK_{v_{i}})}_{i=1}^{p} are mutually disjoint. Hence, by (11) and the claim, we obtain

p ≤ \frac{meas_{2}(B(x, 3r))}{C_{3}r^{2}} ≤ C_{4},  (12)

where, C_{4} is a positive constant independent of r, s and x ∈ J(G). Furthermore, by the definition of l(A), we have diam γ_{1} ⋯ γ_{l(A)−1}(K_{i}) ≤ r. Hence, by the Koebe distortion theorem, there exists a positive constant C_{5}, which is independent of r and x ∈ J(G), such that

∥(γ_{1} ⋯ γ_{l(A)−1})′(z)∥ ≤ C_{5}r,  (13)

for each z ∈ B(K_{i}, \frac{1}{5}ϵ). Hence, by Lemma 5.15, Lemma 3.10, Lemma 4.4, (12) and (13), we obtain

ν(B(x, r)) = ν̃(π−1 ; \overline{ℂ}(B(x, r)) ∩ J̃(f̃))

≤ ∑_{i=1}^{p} ν̃(η^{i}(π−1 ; \overline{ℂ}(B(K_{v_{i}}, \frac{1}{5}ϵ)) ∩ J̃(f̃)))

= ∑_{i=1}^{p} ∫_{π−1 ; \overline{ℂ}(B(K_{v_{i}} , \frac{1}{5} ϵ))∩J̃(f̃)} ∥(γ_{1}^{i} ⋯ γ_{l(R_{i})−1}^{i})′(π\overline{ℂ}(z))∥^{δ} dν̃(z)

≤ C_{4}C_{5}^{δ}r^{δ}.

Similarly, if τ is a t-conformal measure, then by Lemma 5.15, Lemma 4.3, (12), and (13), we obtain

τ(B(x, r)) = τ(B(x, r) ∩ J(G))

≤ ∑_{i=1}^{p} τ(γ_{1}^{i} ⋯ γ_{l(R_{i})−1}^{i}(B(K_{v_{i}}, \frac{1}{5}ϵ) ∩ J(G)))

= ∑_{i=1}^{p} ∫_{B(K_{v_{i}} , \frac{1}{5} ϵ)∩J(G)} ∥(γ_{1}^{i} ⋯ γ_{l(R_{i})−1}^{i})′∥^{t} dτ

≤ C_{4} · C_{5}^{t} · r^{t}.

By Lemma 3.11 and Lemma 3.16, we find that a positive constant C′ exists such that for each r with 0 < r < diam \overline{ℂ} and x ∈ J(G), we have ν(B(x, r)) ≥ C′r^{δ}. Hence, it follows that a positive constant C_{6} exists such that for each r with 0 < r < diam \overline{ℂ} and each x ∈ J(G), we have C_{6}^{−1}r^{δ} ≤ ν(B(x, r)) ≤ C_{6}r^{δ}. Hence, by Proposition 2.2 in [F] and Main Theorem A, we obtain 0 < H^{δ}(J(G)) < ∞ and dim_{H}(J(G)) = \overline{dim}_{B}(J(G)) = δ.

Similarly, if τ is a t-conformal measure, then by Lemma 4.2, τ is t-subconformal. By Lemma 3.16, we find that a positive constant C_{7} exists such that for each r > 0 and x ∈ J(G), we have τ(B(x, r)) ≥ C_{7}r^{t}. Hence, it follows that a positive constant C_{8} exists such that for each r > 0 and x ∈ J(G), we have C_{8}^{−1}r^{t} ≤ τ(B(x, r)) ≤ C_{8}r^{t}. Hence, by Proposition 2.2 in [F], we obtain 0 < H^{t}(J(G)) < ∞ and dim_{H}(J(G)) = t = δ. Then, we find that a positive constant C_{9} exists such that for each x ∈ J(G) and each r > 0, we have C_{9}^{−1}τ(B(x, r)) ≤ ν(B(x, r)) ≤ C_{9}τ(B(x, r)). Hence, by the Besicovitch covering lemma (p294 in [Pe]), we find that ν and τ are absolutely continuous with respect to each other. Hence, we have shown Lemma 5.3.

We now demonstrate Main Theorem B. Proof of Main Theorem B: By Lemma 5.3, we find a positive constant C exists such that for each r with 0 < r < diam \overline{ℂ} and each x ∈ J(G), we have C^{−1}r^{δ} ≤ ν(B(x, r)) ≤ Cr^{δ}. Furthermore, dim_{H}(J(G)) = \overline{dim}_{B}(J(G)) = s(G) = s_{0}(G) = δ. Combining this with Main Theorem A, we see that for each x ∈ \overline{ℂ} \ (A(G) ∪ P (G)), we have dim_{H}(J(G)) = S(x) = T (x) = δ.

By Lemma 5.3 and Proposition 4.13, we obtain ν = \frac{H^{δ}|_{J(G)}}{H^{δ}(J(G))}, ν is a δconformal measure satisfying the separating condition for {f_{1}, ⋯, f_{m}}, and f_{i}^{−1}(J(G))∩f_{j}^{−1}(J(G)) is nowhere dense in f_{j}^{−1}(J(G)) for each (i, j) with i ≠ j.

Let τ be a t-conformal measure. Then, by Lemma 5.3, we have t = δ and τ is absolutely continuous with respect to ν. Since ν satisfies the separating condition for {f_{1}, ⋯, f_{m}}, it follows that τ also satisfies the separating condition for {f_{1}, ⋯, f_{m}}. Combining this with Lemma 4.10-2, we obtain τ = ν.

Hence, we have shown Main Theorem B.

6 Examples

Example 6.1. 1. Let G = ⟨f_{1}, f_{2}⟩ where f_{1}(z) = z^{2} and f_{2}(z) = 2.3(z−3)+ 3. Then, we can see easily that {|z| < 0.9} ⊂ F (G) and G is expanding.

By the corollary 3.17, we get

\overline{dim}_{B}(J(G)) ≤ \frac{log 3}{log 1.8} < 2.

In particular, J(G) has no interior points. In [S3], it was shown that if a finitely generated rational semigroup satisfies the open set condition with an open set U, then the Julia set is equal to the closure of the open set U or has no interior points. Note that the fact that the Julia set of the above semigroup G has no interior points was shown by using analytic quantity only. It appears to be true that G does not satisfy the open set condition.

2. Let G = ⟨\frac{z^{3}}{4} , z^{2} + 8⟩. Then, we can easily see that \{|z| < 2\} ⊂ F (G) and G is expanding. Hence, we have

\overline{dim}_{B}J(G) ≤ \frac{log 5}{log 3} < 2.

In particular, J(G) has no interior points.

Example 6.2. Let p_{1}, p_{2} and p_{3} ∈ ℂ be mutually distinct points such that p_{1}p_{2}p_{3} makes a regular triangle. Let U be the inside part of the regular triangle. Let f_{i}(z) = 2(z − p_{i}) + p_{i} for each i = 1, 2, 3. Let D(x, r) be a Euclidean disk with radius r in U \ ∪_{i=1}^{3}f_{i}^{−1}(\overline{U}), where x denotes the barycenter of the regular triangle p_{1}p_{2}p_{3}. Let g be a polynomial such that J(g) = ∂D(x, r). Let f_{4}(z) = g^{s}(z), where s ∈ ℕ is a large number such that f_{4}^{−1}(U) ⊂ U \ ∪_{i=1}^{3}f_{i}^{−1}(U). Let G = ⟨f_{1}, f_{2}, f_{3}, f_{4}⟩. Then, G satisfies the open set condition with U with respect to {f_{i}}. Furthermore, G is hyperbolic. Hence, G is expanding, by Theorem 2.6 in [S2]. Hence, G satisfies the assumption in Main Theorem B. (Note that J(⟨f_{1}, f_{2}, f_{3}⟩) is the Sierpiński gasket.)

Example 6.3. For any b with 0 < b ≤ 0.1, there exists an a with 0.2 < a ≤ 1 such that G = ⟨a(z − b)^{3} + b, z^{2}⟩ satisfies that (1)G is expanding, (2)G satisfies the open set condition with U = \{z ∈ ℂ | |z − b| < \frac{1}{\sqrt{a}}, |z| > 1\}, (3)J(G) is connected, and (4)J(G) is porous (hence δ = dim_{H}(J(G)) = \overline{dim}_{B}(J(G)) < 2).

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