Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products^{∗}

Hiroki Sumi Department of mathematics, Tokyo Institute of Technology, 2-12-1, Oh-okayama, Meguro-ku, Tokyo, 152-8550, Japan e-mail; sumi@math.titech.ac.jp

June 30, 1999

Abstract

We consider dynamics of sub-hyperbolic and semi-hyperbolic semigroups of rational functions on the Riemann sphere and will show some no wandering domain theorems. The Julia set of a rational semigroup in general may have non-empty interior points. We give a sufficient condition that the Julia set has no interior points. From some information about forward and backward dynamics of the semigroup, we consider when the area of the Julia set is equal to 0 or an upper estimate of the Hausdorff dimension of the Julia set.

For a Riemann surface S, let End(S) denote the set of all holomorphic endomorphisms of S. It is a semigroup with the semigroup operation being composition of maps. 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.

Definition 0.1. 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.

J(G) is backward invariant under G but not forward invariant in general. If G is generated by a compact subset of End(\overline{ℂ}), then J(G) has the backward self-similarity. That is, Lemma 0.2. Let G be a rational semigroup and assume G is generated by a compact subset Λ of End(\overline{ℂ}). Then

∗1991 Mathematical Subject Classification: 30D05, 58F23

J(G) = ⋃_{f∈Λ} f^{−1}(J(G)).

We call this property the backward self-similarity of the Julia set.

Proof. Since J(G) is backward invariant under G, we have

J(G) ⊃ ∪_{f∈Λ}f^{−1}(J(G)).

Suppose there exists a point x ∈ J(G) that does not belong to ∪_{f∈Λ}f^{−1}(J(G)). There exists a neighborhood U of x in \overline{ℂ} such that f(U) ⊂ F (G) for each f ∈ Λ. Take any x′ ∈ U. Let ϵ > 0 be any small number. Since ∪_{f∈Λ}f(x′) is a compact subset of F (G), there exists a number δ_{1} > 0 such that if d(f(x′), y) < δ_{1} for some f ∈ Λ, then d(gf(x′), g(y)) < ϵ for each g ∈ G ∪ {id}. Take δ_{2} > 0 such that if d(x′, y) < δ_{2} then d(f(x′), f(y′)) < δ_{1} for each f ∈ Λ. Then we have that if d(x′, y) < δ_{2}, then d(gf(x′), gf(y′)) < ϵ for each g ∈ G ∪ {id} and each f ∈ Λ. Hence we have x ∈ F (G) and this is a contradiction.

The Julia set of any rational semigroup is a perfect set, backward orbit of any point of the Julia set is dense in the Julia set and the set of repelling fixed points of the semigroup is dense in the Julia set. In general, the Julia set of a rational semigroup may have non-empty interior points. For example, J(⟨z^{2}, 2z⟩) = {|z| ≤ 1}. In fact, in [HM2] it was shown that if G is a finitely generated rational semigroup, then any super attracting fixed point of any element of G does not belong to ∂J(G). Hence we can easily get many examples that the Julia sets have non-empty interior points. For more details about these properties, see [HM1], [HM2], [ZR], [GR], [S1] and [S2]. In this paper we use the notations in [HM1] , [S1] and [S2].

Since the Julia set of a rational semigroup may have non-empty interior points, it is significant for us to get sufficient conditions such that the Julia set has no interior points, to know when the area of the Julia set is equal to 0 or to get an upper estimate of the Hausdorff dimension of the Julia set. We will try that using various information about forward dynamics of the semigroup in the Fatou set or backward dynamics of the semigroup in the Julia set.

In the section 1 of this paper we will define sub-hyperbolic and semihyperbolic rational semigroups and show no wandering domain theorems. In particular, we will see that if G is a finitely generated sub-hyperbolic or semi-hyperbolic rational semigroup, then there exists an attractor in the Fatou set for G(Theorem 1.34). By using these theorems, we can show the continuity of the Julia set with respect to the perturbation of the generators(Corollary 1.39).

In Section 2, we will consider the skew products of rational functions or dynamics on \overline{ℂ}-fibrations. The “Julia set” of any skew product is defined to be the closure of the union of the fibrewise Julia sets. We will define hyperbolicity and semi-hyperbolicity. We will show that if a skew product is semi-hyperbolic, then the Julia set is equal to the union of the fibrewise Julia sets and the skew product has the contraction property with respect to the backward dynamics along fibres(Theorem 2.13). The results in section 2 are generalized to those of version of dynamics on \overline{ℂ}-fibrations.

In section 3, we will consider necessary and sufficient conditions to be semi-hyperbolic(Theorem 3.1, Theorem 3.5). We will show that any subhyperbolic semigroup without any superattracting fixed point of any element of the semigroup in the Julia set is semi-hyperbolic(Theorem 3.7).

In section 4, we will show that if a finitely generated rational semigroup G is semi-hyperbolic and satisfies the open set condition with an open set O such that ♯(∂O ∩ J(G)) < ∞ , then 2-dimensional Lebesgue measure of the Julia set is equal to 0(Theorem 4.4).

In section 5, we will consider constructing δ-subconformal measures. If a rational semigroup has at most countably many elements and the δ-Poincaré series converges, then we can construct δ-subconformal measures. We will see that if G is a finitely generated semi-hyperbolic rational semigroup, then the Hausdorff dimension of the Julia set is less than the exponent δ(Theorem 5.6, Theorem 5.7). To show those results, the contracting property of backward dynamics will be used.

Note. This paper belongs to a part of the author’s thesis([S3]) which was written under supervision of Prof. Shigehiro Ushiki.

ACKNOWLEDGEMENT. The author would like to express his gratitude to Prof. Shigehiro Ushiki for many valuable discussions and advices.

The author would also like to express his gratitude to Prof. Manfred Denker, Hartje Kriete and Stefan Heinemann for their kind hospitality during my stay in Goettingen and many valuable discussions.

The author is grateful to Prof. Olivier Sester for an interesting suggestion about hyperbolicity on skew products in a conference held at Berlin.

1 No Wandering Domain

Definition 1.1. Let G be a rational semigroup. We set

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

We call P (G) the post critical set of G. We say that G is hyperbolic if P (G) ⊂ F (G). Also we say that G is sub-hyperbolic if ♯{P (G) ∩ J(G)} < ∞ and P (G) ∩ F (G) is a compact set.

We denote by B(x, ϵ) a ball of center x and radius ϵ in the spherical metric. We denote by D(x, ϵ) a ball of center x ∈ ℂ and radius ϵ in the Euclidian metric. Also for any hyperboplic manifold M we denote by H(x, ϵ) a ball of center x ∈ M and radius ϵ in the hyperbolic metric. For any rational map g, we denote by B_{g}(x, ϵ) a connected component of g^{−1}(B(x, ϵ)). For each open set U in \overline{ℂ} and each rational map g, we denote by c(U, g) the set of all connected components of g^{−1}(U). Note that if g is a polynomial and U = D(x, r) then any element of c(U, g) is simply connected by the maximal principle.

For each set A in \overline{ℂ}, we denote by A^{i} the set of all interior points of A.

Definition 1.2. Let G be a rational semigroup and A a set in \overline{ℂ}. We set G(A) = ∪_{g∈G}g(A) and G^{−1}(A) = ∪_{g∈G}g^{−1}(A).

We can show the following Lemma immediately.

Lemma 1.3. Let G be a rational semigroup. Assume that {f_{λ}}_{λ∈Λ} is a generator system of G. Then we have

⋃_{g∈G}{critical values of g} = ⋃_{λ∈Λ}(G ∪ {Id})({critical values of f_{λ}}).

Definition 1.4. Let G be a rational semigroup and N a positive integer. We set

SH_{N}(G)
= \{x ∈ \overline{ℂ}|∃δ(x) > 0, ∀g ∈ G, ∀B_{g}(x, δ(x)), deg(g : B_{g}(x, δ) → B(x, δ)) ≤ N\}

and UH(G) = \overline{ℂ} \ (∪_{N∈ℕ}SH_{N}(G)).

Remark 1. By definition, SH_{N}(G) is an open set in \overline{ℂ} and g^{−1}(SH_{N}(G)) ⊂ SH_{N}(G) for each g ∈ G. Also UH(G) is a compact set and g(UH(G)) ⊂ UH(G) for each g ∈ G. For each rational map g with deg(g) ≤ 2, any parabolic or attracting periodic point of g belongs to UH(G).

Definition 1.5. Let G be a rational semigroup. We say that G is semihyperbolic (resp. weakly semi-hyperbolic) if there exists a positive integer N such that J(G) ⊂ SH_{N}(G)(resp.∂J(G) ⊂ SH_{N}(G)).

Remark 2. 1. If G is semi-hyperbolic and N = 1, then G is hyperbolic.

2. If G is hyperbolic, then G is semi-hyperbolic.

3. For a rational map f with the degree at least two, ⟨f⟩ is semi-hyperbolic if and only if f has no parabolic orbits and each critical point in the Julia set is non-recurrent([CJY], [Y]). If ⟨f⟩ is semi-hyperbolic, then there are neither indifferent cycles, Siegel disks nor Hermann rings.

Definition 1.6. Let V be a domain in \overline{ℂ} and E a compact subset of V. We set

mod (E, V) = sup{ mod A},

where the supremum is taken over all annulus A such that E lies in a compact component of V \ A.

Lemma 1.7 ([CJY]). 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). Here we can take C = C(N, r) independent of f.

Corollary 1.8. For any positive integer N and real number r with 0 < r < 1, there exist constants r_{1} and r_{2} with 0 < r_{1} ≤ r_{2} < 1 depending only on r, N such that if f : D(0, 1) → D(0, 1) is a proper holomorphic map with deg(f) = N and f(0) = 0, then

D(0, r_{1}) ⊂ W ⊂ D(0, r_{2})

where W is the connected component of f^{−1}(D(0, r)) containing 0.

Corollary 1.9 ([Y]). Let V be a simply connected domain in ℂ, 0 ∈ V, f : V → D(0, 1) be a proper holomorphic map of degree N and f(0) = 0, W be the component of f^{−1}(D(0, r)) containing 0, 0 < r < 1. Then there exists a constant K depending only on r and N, not depending on V and f, so that

|\frac{x}{y}| ≤ K

for all x, y ∈ ∂W.

Proof. We will follow Y.Yin’s proof([Y]). Let g : V → D(0, 1) be the univalent function such that g(0) = 0. ¿From Corollary 1.8,

r_{1} ≤ |g(x)| ≤ r_{2}

for all x ∈ ∂W. Applying the Koebe distortion theorem, we have that

|(g^{−1})′(0)| · \frac{r_{1}}{(1 + r_{1})^{2}} ≤ |x| ≤ |(g^{−1})′(0)| · \frac{r_{2}}{(1 − r_{2})^{2}} .

Then

|\frac{x}{y}| ≤ \frac{r_{2}(1 + r_{1})^{2}}{r_{1}(1 − r_{2})^{2}} =: K.

K is a constant depending only on N and r.

Lemma 1.10. 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 satisfying that for each connected component U of f^{−1}(D(0, r)),

diam_{S} U ≤ C,

where we denote by diam_{S} the spherical diameter. Also we have C(N, r) → 0 as r → 0.

Proof. Let r be a number with 0 < r < 1. Let U be a connected component of f^{−1}(D(0, r)) and V′ be the connected component of \overline{ℂ} \ \overline{V} containing K. Since V is connected, V′ is simply connected. Let U′ be the connected component of \overline{ℂ} \ \overline{U} containing V′. Since U′ is also simply connected and \overline{V′} ⊂ U′, we have that there exists a connected component of U′ \ \overline{V′} which is a ring domain.

There exists a sequence (r_{j})_{j=0}^{n} of real numbers with r_{0} = r < r_{1} < ⋯ < r_{n} = 1 such that there exist no critical values of f in D(0, r_{j+1})\\overline{D(0, r_{j})} for j = 0, …, n − 1. For each i = 0, …, n, let U′′_{i} be the connected component of f^{−1}(D(0, r_{i})) containing U and let U′_{i} be the connected component of \overline{ℂ} \ \overline{U′′_{i}} containing V′. Then we have

U′′_{0} = U ⊂ U′′_{1} ⊂ ⋯ ⊂ U′′_{n} = V and

U′_{0} = U′ ⊃ U′_{1} ⊃ ⋯ ⊃ U′_{n} = V′.

By the construction, f : U′′_{i+1} \ \overline{U′′_{i}} → D(0, r_{i+1}) \ \overline{D(0, r_{i})} is a proper map for i = 0, …, n − 1. Since there exist no critical values of f in D(0, r_{i+1}) \ \overline{D(0, r_{i})}, each connected component of U′′_{i+1} \ \overline{U′′_{i}} is a ring domain.

Now we claim that for each i = 0, …, n − 1, there exists a connencted component of U′′_{i+1} \ \overline{U′′_{i}} which is included in U′_{i} \ \overline{U′_{i+1}}. We will show that. Since ∂U′_{i} ⊂ U′′_{i+1}, there exists a ring domain R_{i} in U′′_{i+1} \ \overline{U′′_{i}} such that ∂U′_{i} is a connected component of ∂R_{i}. Let R′_{i} be the connected component of U′′_{i+1} \ \overline{U′′_{i}} containing R_{i}. Since

∂(U′_{i} \ \overline{U′_{i+1}}) = ∂U′_{i} ∪ ∂U′_{i+1} ⊂ ∂U′′_{i} ∪ ∂U′′_{i+1},

we have R′_{i} ∩ ∂(U′_{i} \ \overline{U′_{i+1}}) = ∅. Hence R′_{i} ⊂ U′_{i} \ \overline{U′_{i+1}} and we have proved the above claim.

From the above claim, we get

mod (\overline{U′_{i+1}}, U′_{i}) ≥ \frac{1}{2πN} log \frac{r_{i+1}}{r_{i}} , for i = 0, …, n − 1.

It follows that

mod (\overline{V′}, U′) ≥ ∑_{i=0}^{n−1} mod (U′_{i} \ \overline{U′_{i+1}})
≥ \frac{1}{2πN} log \frac{1}{r}.

On the other hand, by Lemma 6.1 in p34 in [LV], we have

mod (\overline{V′}, U′) ≤ \frac{π^{2}}{2C_{1}^{2}} ,

where C_{1} = min{a, diam_{S} U}. Hence the statement of our lemma holds.

Lemma 1.11. Let V and W be simply connected domains in \overline{ℂ}. Suppose that \overline{W} ⊂ V and mod (\overline{W} , V) > c > 0. Then there exists a constant 0 < λ < 1 depending only on c such that

\frac{diam W}{diam V} ≤ λ,

here by “diam” we mean the spherical diameter.

Proof. We can assume that 0 ∈ W and diam V = d(0, 1) where d is the spherical metric. Let g : D(0, 1) → V be the Riemann map such that g(0) = 0. By Theorem 2.4 in [M], there exists a constant c_{1} depending only on c such that

diam_{H}(g^{−1}(W)) ≤ c_{1},

where we denote by diam_{H} the diameter with respect to the hyperbolic metric in D(0, 1). Since diam V = d(0, 1), by the Koebe distortion theorem, we have that there exists a constant c_{2} not depending on V and W such that |g′(0)| ≤ c_{2}. Using the Koebe distortion theorem again, we see that there exists a constant c_{3} depending only on c such that for each z ∈ g^{−1}(W), |g(z)| ≤ c_{3}. Hence there exists a constant 0 < c_{4} < d(0, 1) depending only on c such that diam W ≤ c_{4}.

Lemma 1.12. Let G = ⟨f_{1}, …, f_{m}⟩ be a finitely generated rational semigroup. Let y be a point of \overline{ℂ} \ UH(G). If there exists a neighborhood W of y such that \overline{ℂ} \ G^{−1}(W) contains a continuum, then there exists a neighborhood W_{1} of y such that for each simply connected open neighborhood V of y included in W_{1} and for each g ∈ G, each element of c(V, g) is simply connected.

Proof. For each j = 1, …, m, let C_{j} be the set of all critical points of f_{j}. By Lemma 1.10, there exists a δ > 0 such that for each g ∈ G, each element of c(B(y, δ), g) does not contain any two different points of C_{j}, j = 1, … m. Then for any simply connected open neighborhood V of y included in B(y, δ) and for any g ∈ G, each element of c(V, g) is simply connected.

Lemma 1.13. Let G be a rational semigroup and N a positive integer. Then for each g ∈ G, any critical point c of g does not belong to SH_{N}(G) ∩ \overline{(G ∪ \{id\})(g(c))}.

Proof. Assume that there exists a critical point c of an element g ∈ G such that c ∈ SH_{N}(G) ∩ \overline{(G ∪ \{id\})(g(c))}. Then there exists a sequence (g_{n}) in G so that g_{n}g(c) → c.

There exists a positive number ϵ such that B(c, ϵ) ⊂ SH_{N}(G). Since g_{n}g(c) → c, we can construct a sequence (n_{j}) and a sequence (B_{j}) so that for each j, B_{j} is a connected component of ((g_{n_{1}}g)(g_{n_{2}}g) ⋯ (g_{n_{j}}g))^{−1}(B(c, ϵ)) and c ∈ B_{j}, which contradicts that c ∈ SH_{N}(G).

Lemma 1.14. Let g be a rational map with deg(g) ≥ 2 and N a positive integer. Assume that x ∈ J(⟨g⟩) ∩ SH_{N}(⟨g⟩). Then x belongs to neither boundaries of Siegel disks, boundaries of Hermann rings nor indifferent cycles.

Proof. By Theorem 1 and Corollary in [Ma] and Lemma 1.13, we can show the statement immediately.

Definition 1.15. Let G be a rational semigroup and U a component of F (G). For every element g of G, we denote by U_{g} the connected component of F (G) containing g(U). We say that U is a wandering domain if {U_{g}} is infinite.

Remark 3. In [HM1], A.Hinkkanen and G.J.Martin showed that there exists an infinitely generated polynomial semigroup which has a wandering domain.

Lemma 1.16. Let G be a rational semigroup which contains an element with the degree at least two. Let x be a point of F (G) and assume that there exists a point y ∈ ∂J(G) and a sequence (g_{n}) of elements of G such that g_{n}(x) → y. Then we have y ∈ P (G) ∩ ∂J(G).

Proof. We can assume that ♯P (G) ≥ 3. Suppose y ∈ \overline{ℂ} \ P (G). Let δ be a number so that \overline{B(y, δ)} ⊂ \overline{ℂ}\P (G). We can assume that for each n, g_{n}(x) ∈ B(y, δ). For each n, there exists an analytic inverse branch α_{n} of g_{n} in U such that α_{n}(g_{n}(x)) = x. Since ♯P (G) ≥ 3, we have {α_{n}}is normal in U. Hence if we take an ϵ small enough,

diam α_{n}(B(y, ϵδ)) < d(x, J(G)), for each n.

But x ∈ α_{n}(B(y, ϵδ)) for large n and α_{n}(B(y, ϵδ)) ∩ J(G) ≠ ∅ because J(G) is backward invariant under G. This is a contradiction.

Corollary 1.17. Let G be a rational semigroup which contains an element with the degree at least two. If P (G) ∩ ∂J(G) = ∅, then for each x ∈ F (G), \overline{G(x)} \ F (G) and there is no wandering domain.

Lemma 1.18. Let G be a polynomial semigroup, N a positive integer and y a point in ∂J(G) ∩ ℂ. Assume that there exists an open neighborhood U of y such that U ⊂ SH_{N}(G) and ♯(\overline{ℂ} \ G^{−1}(U)) ≥ 3. Then for each x ∈ F (G), \overline{G(x)} ⊂ \overline{ℂ} \ \{y\}.

Proof. We can assume that ∞ ∈ \overline{ℂ} \ G^{−1}(U). Suppose that there exists a point x ∈ F (G) and a sequence (g_{n}) in G such that g_{n}(x) → y as n → ∞. Let δ be a positive number so that for each g ∈ G,

deg(g : V → D(y, δ)) ≤ N,

for each V ∈ c(D(y, δ), g). For any r with 0 < r ≤ δ there exists a positive integer n(r) such that for each integer n with n ≥ n(r), g_{n}(x) ∈ D(y, r). Let D_{g_{n}}(y, r) be the connected component of g_{n}^{−1}(D(y, r)) containing x. For each n with n ≥ n(r), there exists a conformal map φ_{n} from D(0, 1) onto D_{g_{n}}(y, δ) such that φ_{n}(0) = x. From Lemma 1.10, there exists a constant C(r) with C(r) → 0 as r → 0 such that for each integer n with n ≥ n(r),

diam φ_{n}^{−1}(D_{g_{n}}(y, r)) ≤ C(r).

Since ♯(\overline{ℂ} \ G^{−1}(U)) ≥ 3, the family \{φ_{n}\} is normal in D(0, 1). Hence if r is sufficiently small, then for each integer n with n ≥ n(r),

diam_{S}D_{g_{n}}(y, r) < d(J(G), x),

where we denote by diam_{S} the spherical diameter and by d the spherical distance. On the other hand, since J(G) is backward invariant under G and y ∈ J(G), we have that for each n with n ≥ n(r), D_{g_{n}}(y, r)∩J(G) ≠ ∅. This is a contradiction. Therefore we have for each x ∈ F (G), \overline{G(x)} ⊂ \overline{ℂ}\\{y\}.

Lemma 1.19. Let G be a polynomial semigroup. Assume that there exists a point x ∈ F (G), a point y ∈ ∂J(G) and a sequence (g_{n}) in G such that g_{n}(x) → y as n → ∞. Then at least one of the following holds.

1. UH(G) = ∅ and each element of G is a Möbius transformation. For each z ∈ F (G), y ∈ \overline{G(z)}.

2. ♯(UH(G)) = 1or 2, UH(G) ⊂ J(G) and UH(G) ∩ ∂J(G) ≠ ∅. For each z ∈ F (G), y ∈ \overline{G(z)}.

3. y ∈ UH(G).

Proof. Suppose that ♯(UH(G)) ≥ 3. From Lemma 1.18, we have y ∈ UH(G).

Suppose there exists a point z ∈ F (G) such that \overline{G(z)} ⊂ \overline{ℂ} \ \{y\}. Then there exists a neighborhood V of z such that G(V) ⊂ \overline{ℂ} \ \{y\}. By Lemma 1.18, y ∈ UH(G).

Now we consider the case ♯(UH(G)) = 1 or 2. Then ∞ ∈ UH(G). If ∞ ∈ F (G), then since G(∞) = {∞}, from Lemma 1.18 the condition 3. holds. Now suppose ∞ ∈ J(G). There exists an element g ∈ G with the degree at least two. From Corollary 1.14, g has no Siegel disks. Let z be a point in F (G). Since F (G) ⊂ F (⟨g⟩), z ∈ F (⟨g⟩). From no wandering domain theorem and the fact that g has no Siegel disks, there exists an attracting or parabolic periodic point ζ ∈ \overline{F (G)} of g and a sequence (n_{j}) of positive integers such that g^{n_{j}}(z) → ζ. We have ζ ∈ UH(G). If ζ ∈ ∂J(G), then the condition 2. holds. If ζ ∈ F (G), then since G is a polynomial semigroup, we have G({ζ}) = {ζ} ⊂ F (G) and it implies y ∈ UH(G) from Lemma 1.18. Hence the condition 3. holds.

Finally we consider the case UH(G) = ∅. Assume there exists an element h ∈ G with the degree at least two. Since F (G) ≠ ∅, we have F (⟨g⟩) ≠ ∅. By the no wandering domain theorem, g has (super)attracting cycles, parbolic cycles, Siegel disks or Hermann rings. Since UH(G) = ∅, this is a contradiction.

Theorem 1.20. Let G be a rational semigroup containing an element with the degree at least two and U a connected component. Assume that there exists a sequence (g_{n}) of elements of G such that U_{g_{n}} ∩ U_{g_{m}} = ∅ if n ≠ m ( in pariticular, U is a wandering domain). Then there exists a subsequence (g_{n_{j}}) of (g_{n}) and a point y ∈ P (G) ∩ ∂J(G) such that (g_{n_{j}}) converges to y locally uniformly on U.

Proof. By the method in the proof of Theorem 2.2.3 in [S1], we can show that there exists a subsequence (g_{n_{j}}) of (g_{n}) and a point y ∈ ∂J(G) such that (g_{n_{j}}) converges to y locally uniformly on U. Hence the statement of our theorem holds from Lemma 1.16.

Theorem 1.21. Let G be a polynomial semigroup and U a connected component of F (G). Assume that there exists a sequence (g_{n}) of elements of G such that U_{g_{n}} ∩U_{g_{m}} = ∅ if n ≠ m ( in pariticular, U is a wandering domain). Then at least one of the following holds.

1. UH(G) = ∅ and each element of G is a Möbius transformation. For each z ∈ F (G), \overline{G(z)} ∩ ∂J(G) ≠ ∅.

2. ♯(UH(G)) = 1 or 2, UH(G) ⊂ J(G) and UH(G) ∩ ∂J(G) ≠ ∅. For each z ∈ F (G), \overline{G(z)} ∩ ∂J(G) ≠ ∅.

3. There exists a subsequence (g_{n_{j}}) of (g_{n}) and a point y ∈ UH(G) ∩ ∂J(G) such that (g_{n_{j}}) converges to y locally uniformly on U.

Proof. Using Lemma 1.19, we can show the statement in the same way as the proof of Theorem 1.20.

By Lemma 1.10 and using the method of the proof in Lemma 1.18, we can show the next lemma immediately.

Lemma 1.22. Let G be a rational semigroup and y a point of ∂J(G) \ UH(G). Assume that there exists an open neighborhood U of y such that \overline{ℂ} \ G^{−1}(U) contains a continuum K. Then for each x ∈ F (G), \overline{G(x)} ⊂ \overline{ℂ} \ \{y\}.

Lemma 1.23. Let G be a rational semigroup. Assume that there exists a point x ∈ F (G), a point y ∈ ∂J(G) and a suquence (g_{n}) in G such that g_{n}(x) → y as n → ∞. Then at least one of the following holds.

1. UH(G) = ∅ and each element of G is a Möbius transformation. For each z ∈ F (G), y ∈ \overline{G(z)}.

2. UH(G) is totally disconnected, UH(G) ⊂ J(G) and UH(G)∩∂J(G) ≠ ∅. For each z ∈ F (G), y ∈ \overline{G(z)}.

3. y ∈ UH(G).

Proof. Suppose UH(G) is empty. Then we can show that each element of G is a Möbius transformation in the same way as the proof of Lemma 1.19.

Suppose there exists a point z ∈ F (G) such that \overline{G(z)} ⊂ \overline{ℂ} \ \{y\}. Then there exists a neighborhood V of z such that G(V) ⊂ \overline{ℂ} \ \{y\}. By Lemma 1.22, y ∈ UH(G).

Suppose UH(G) ∩ F (G) ≠ ∅. Let z ∈ UH(G) ∩ F (G). If \overline{G(z)} ⊂ \overline{ℂ} \ \{y\}, then by the previous arguments, y ∈ UH(G). If y ∈ \overline{G(z)}, we have also y ∈ UH(G).

If UH(G) contains a continuum, then from Lemma 1.22, we have y ∈ UH(G).

Suppose that ∅ ≠ UH(G) ⊂ J(G) and UH(G) is totally disconnected. There exists an element g ∈ G of degree at least two. Since UH(G) is totally disconnected and F (G) ≠ ∅, by no wandering domain theorem we can show that g has an (super) attracting or parabolic periodic point ζ in ∂J(G). We have ζ ∈ UH(G).

By Lemma 1.23, we can show the next result in the same way as the proof of Theorem 1.20.

Theorem 1.24. Let G be a rational semigroup and U a connected component of F (G). Assume that there exists a sequence (g_{n}) of elements of G such that U_{g_{n}} ∩ U_{g_{m}} = ∅ if n ≠ m ( in pariticular, U is a wandering domain). Then at least one of the following holds.

1. UH(G) = ∅ and each element of G is a Möbius transformation. For each z ∈ F (G), \overline{G(z)} ∩ ∂J(G) ≠ ∅.

2. UH(G) is totally disconnected, UH(G) ⊂ J(G) and UH(G)∩∂J(G) ≠ ∅. For each z ∈ F (G), \overline{G(z)} ∩ ∂J(G) ≠ ∅.

3. There exists a subsequence (g_{n_{j}}) of (g_{n}) and a point y ∈ UH(G) ∩ ∂J(G) such that (g_{n_{j}}) converges to y locally uniformly on U.

By Lemma 1.22, we can show the next result immediately.

Theorem 1.25. Let G be a rational semigroup. Assume that G is weakly semi-hyperbolic and there is a point z ∈ F (G) such that the closure of the G-orbit \overline{G(z)} is included in F (G). Then for each x ∈ F (G), \overline{G(x)} ⊂ F (G) and there is no wandering domain.

Next theorem follows from Lemma 1.23.

Theorem 1.26. Let G be a rational semigroup containing an element g ∈ G with deg(g) ≥ 2. Assume that G is weakly semi-hyperbolic. If F (G) ≠ ∅, then for each x ∈ F (G), \overline{G(x)} ⊂ F (G) and there is no wandering domain.

Definition 1.27. Let G be a rational semigroup. We set

A_{0}(G) = \overline{G(\{z ∈ \overline{ℂ} | ∃g ∈ G with deg(g) ≥ 2, g(x) = x and |g′(x)| < 1.\})},

Ã_{0}(G) = \overline{G(\{z ∈ F (G) | ∃g ∈ G with deg(g) ≥ 2, g(x) = x and |g′(x)| < 1.\})},

A(G) = \overline{G(\{z ∈ \overline{ℂ} | ∃g ∈ G, g(x) = x and |g′(x)| < 1.\})},
Ã(G) = \overline{G(\{z ∈ F (G) | ∃g ∈ G, g(x) = x and |g′(x)| < 1.\})},

where the closure in the definition of Ã_{0}(G) and Ã(G) is considered in \overline{ℂ}.

Remark 4. By definition, A_{0}(G) ⊂ A(G)∩P (G). For each g ∈ G, g(A_{0}(G)) ⊂ A_{0}(G) and g(A(G)) ⊂ A(G). We have also similar statements for Ã_{0}(G) and Ã(G).

Lemma 1.28. Let G be a rational semigroup. If Ã_{0}(G) is a non-empty compact subset of F (G), then

∅ ≠ Ã_{0}(G) = Ã(G) ⊂ P (G) ∩ F (G).

Proof. Let g be any Möbius transformation in G and x ∈ \overline{ℂ} a fixed point of g with |g′(x)| < 1. Since g(Ã_{0}(G)) ⊂ Ã_{0}(G) ∩ F (G) and Ã_{0}(G) ≠ ∅, we have that x ∈ Ã_{0}(G). Therefore the statement follows.

Lemma 1.29. Let G be a rational semigroup containing an element with the degree at least two. Assume that G is semi-hyperbolic and F (G) ≠ ∅. Then

∅ ≠ A_{0}(G) = Ã_{0}(G) = A(G) = Ã(G) ⊂ F (G).

Proof. Let g ∈ G be an element with the degree at least two. Since F (G) ≠ ∅, the element g has a (super)attracting periodic point x in \overline{F (G)}. By Remark 1, we have that A_{0}(G) ⊂ F (G). Hence the statement follows from the proof of Lemma 1.28.

Lemma 1.30. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated rational semigroup. Assume that each element of G with the degree at least two has neither Siegel disks nor Hermann rings and each element of Aut \overline{ℂ}∩G(if this is not empty) is loxodromic. Also assume that ♯J(G) ≥ 3. Let U_{1}, …, U_{s} be some connected components of F (G) and K a non-empty compact subset of V = ∪_{j=1}^{s}U_{j} such that U_{j} ∩ K ≠ ∅ for each j = 1, …, s and g(K) ⊂ K for each g ∈ G. Then for each compact subset L of V there exists a constant c with c > 0 and a constant λ with 0 < λ < 1 such that

1. sup{∥(f_{i_{n}} ⋯ f_{i_{1}})′(z)∥ | z ∈ L, (i_{n}, …, i_{1}) ∈ {1, …, m}^{n}} ≤ cλ^{n}, where we denote by ∥ · ∥ the norm of the derivative of with respect to the hyperbolic metric on V.

2. sup{d(f_{i_{n}} ⋯ f_{i_{1}}(z), K) | z ∈ L, (i_{n}, …, i_{1}) ∈ {1, …, m}^{n}} ≤ cλ^{n}, where we denote by d the spherical metric.

Proof. Let a be a large positive number. For each j = 1, …, s, let K_{j} be the compact a-neighborhood of K ∩ U_{j} in U_{j} with respect to the distance induced by the hyperbolic metric in U_{j}. We set K_{0} = ∪_{j=1}^{s}K_{j}. Then for each g ∈ G, g(K_{0}) ⊂ K_{0}. If a is large enough, we have that L ⊂ K_{0}.

We claim that there exists a constant c > 0 and a constant λ < 1 such that

sup{∥(f_{i_{n}} ⋯ f_{i_{1}})′(z)∥ | z ∈ L, (i_{n}, …, i_{1}) ∈ {1, …, m}^{n}} ≤ cλ^{n}, (1)

where we denote by ∥ · ∥ the norm of the derivative of with respect to the hyperbolic metric on V. To show the claim, let z be a point of K_{j} and (i_{s+1}, …, i_{1}) an element of {1, …, m}^{s+1}. Then there exists an integer t with 1 ≤ t ≤ s such that (f_{i_{s+1}} ⋯ f_{i_{t+1}})(U_{j_{t}}) ⊂ U_{j_{t}}, where U_{j_{t}} is the component of V containing (f_{i_{t}} ⋯ f_{i_{1}})(U_{j}). From the assumption, we have that for each x ∈ K_{j_{t}}, ∥(f_{i_{s+1}} ⋯ f_{i_{t+1}})′(x)∥ < 1. Hence

∥(f_{i_{s+1}} ⋯ f_{i_{1}})′(z)∥ < 1.

Therefore the claim holds.

From the above claim, we can show the statement of our lemma immediately.

Definition 1.31. Let G be a rational semigroup and U a open set in \overline{ℂ}. We say that a non-empty compact subset K of U is an attractor in U for G if g(K) ⊂ K for each g ∈ G and for any open neighborhood V of K in U and each z ∈ U, g(z) ∈ U for all but finitely many g ∈ G.

Lemma 1.32. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated rational semigroup and E a finite subset of \overline{ℂ}. Assume that each x ∈ E is not a non-repelling fixed point of any element of G. Then for any M > 0, there exists a positive integer n_{0} such that for any integer n with n ≥ n_{0} if z, f_{w_{1}}(z), f_{w_{2}}f_{w_{1}}(z), …, (f_{w_{n−1}} ⋯ f_{w_{1}})(z) and (f_{w_{n}} ⋯ f_{w_{1}})(z) belong to E and |(f_{w_{n}} ⋯ f_{w_{1}})′(z)| ≠ 0, then |(f_{w_{n}} ⋯ f_{w_{1}})′(z)| > M.

Proof. We will show the statement by induction on ♯E. When ♯E = 1, it easy to see that the statement holds. Now assume that for each finite subset E of \overline{ℂ} with ♯E ≤ s the statement holds. Let E′ be a finite subset of \overline{ℂ} with ♯E′ = s + 1 and assume that each x ∈ E′ is not a non-repelling fixed point of any element of G. Take a number M_{0} so that

M_{0}(inf{|(f_{j})′(ζ)| | ζ ∈ E′, (f_{j})′(ζ) ≠ 0, j = 1, …, m.})^{2} > 1.

From the hypothesis of the induction, there exists a positive integer n_{0} such that for any subset E of E′ with E ≠ E′ and for any integer n with n ≥ n_{0}, if x, f_{w_{1}}(x), f_{w_{2}}f_{w_{1}}(x), …, (f_{w_{n−1}} ⋯ f_{w_{1}})(x) and (f_{w_{n}} ⋯ f_{w_{1}})(x) belong to E and |(f_{w_{n}} ⋯ f_{w_{1}})′(x)| ≠ 0, then |(f_{w_{n}} ⋯ f_{w_{1}})′(x)| > M_{0}. For each y ∈ E and postive integer t with t ≤ n_{0} + 1, we set

G_{y,t} = {g ∈ G | g(y) = y, g: a product of t generators }.

Then we have that ♯G_{y,t} < ∞ and for each g ∈ G_{y,t}, y is a repelling fixed point of g.

Now assume that z, f_{w_{1}}(z), f_{w_{2}}f_{w_{1}}(z), …, (f_{w_{n−1}} ⋯ f_{w_{1}})(z) and (f_{w_{n}} ⋯ f_{w_{1}})(z) belong to E′, (f_{w_{n}} ⋯ f_{w_{1}})(z) = z, (f_{w_{n}} ⋯ f_{w_{1}})′(z) ≠ 0 and (f_{w_{j}} ⋯ f_{w_{1}})(z) ≠ z for each j = 1, …, n − 1. If n ≤ n_{0} + 1, we have

|(f_{w_{n}} ⋯ f_{w_{1}})′(z)| > inf{|g′(z)| | g ∈ G_{z,t}, 1 ≤ t ≤ n_{0} + 1} > 1.

If n ≥ n_{0} + 2, then we have

|(f_{w_{n}} ⋯ f_{w_{1}})′(z)| > M_{0}(inf{|(f_{j})′(ζ)| | ζ ∈ E′, f′_{j}(ζ) ≠ 0, j = 1, …, m.})^{2} > 1.

From these results, we can show that for any M > 0, there exits a positive integer n_{1} such that for any integer u with u ≥ n_{1} if z, f_{w_{1}}(z), f_{w_{2}}f_{w_{1}}(z), …, (f_{w_{u−1}} ⋯ f_{w_{1}})(z) and (f_{w_{u}} ⋯ f_{w_{1}})(z) belong to E′ and |(f_{w_{u}} ⋯ f_{w_{1}})′(z)| ≠ 0, then

(f_{w_{u}} ⋯ f_{w_{1}})′(z)| > M.

Hence we have completed the induction.

Lemma 1.33. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated rational semigroup and E a finite subset of \overline{ℂ}. Assume that each x ∈ E is not any non-repelling fixed point of any element of G. Then there exists an open neighborhood V of E in \overline{ℂ} such that for each z ∈ V, if there exists a word w = (w_{1}, w_{2}, …) ∈ {1, …, m}^{ℕ} satifying that:

1. for each n, (f_{w_{n}} ⋯ f_{w_{1}})(z) ∈ V,

2. (f_{w_{n}} ⋯ f_{w_{1}}(z)) accumulates only in E and

3. for each n, (f_{w_{n}} ⋯ f_{w_{1}})(ζ) ∈ E and (f_{w_{n}} ⋯ f_{w_{1}})′(ζ) ≠ 0 where ζ is the closest point to z in E,

then z is equal to the point ζ ∈ E.

Proof. Let ϵ be a small number so that B(x, ϵ) ∩ B(y, ϵ) = ∅ if x, y ∈ E and x ≠ y. Take an ϵ smaller, if necesarry, so that if z_{0} ∈ E and f′_{j}(z_{0}) ≠ 0 for some j, then f_{j}|_{B(z_{0},ϵ)} is injective. We set V = ∪_{z∈E}B(z, ϵ).

Let z ∈ V be a point. Assume that there exists a word w = (w_{1}, w_{2} …) ∈ {1, …, m}^{ℕ} satisfying the conditions 1, 2 and 3. We set α_{n} = f_{w_{n}}f_{w_{n−1}} ⋯ f_{w_{1}}. From the conditions 2 and 3, there exists a point a ∈ E and a sequence (n_{j}) such that α_{n_{j}}(z) → a as j → ∞ and a_{n_{j}}(ζ) = a for each j. By lemma 1.32, we have |(α_{n})′(ζ)| → ∞ as n → ∞. Hence by the Koebe distortion theorem, there exists a number η > 0 such that for each positive integer j, there exists an analytic inverse branch β_{j} of α_{n_{j}} on B(a, η) so that β_{j}(a) = ζ and β_{j}(B(a, η)) ⊂ V and diam β_{t}(B(a, η)) → 0 as t → ∞.

We set y_{j} = β_{j}(α_{n_{j}}(z)) for each large j. We claim that for each integer l with 0 ≤ l ≤ n_{j} − 1, if (f_{w_{l+1}}f_{w_{l}} ⋯ f_{w_{1}})(y_{j}) = (f_{w_{l+1}}f_{w_{l}} ⋯ f_{w_{1}})(z), then (f_{w_{l}}f_{w_{l−1}} ⋯ f_{w_{1}})(y_{j}) = (f_{w_{l}}f_{w_{l−1}} ⋯ f_{w_{1}})(z). Let us show the claim above. Assume that (f_{w_{l+1}}f_{w_{l}} ⋯ f_{w_{1}})(y_{j}) = (f_{w_{l+1}}f_{w_{l}} ⋯ f_{w_{1}})(z). We have that

f_{w_{l}}f_{w_{l−1}} ⋯ f_{w_{1}} ∘ β_{j} : B(a, η) → \overline{ℂ}

is an analytic inverse branch of f_{w_{n_{j}}} f_{w_{n_{j} −1}} ⋯ f_{w_{l+1}} satisfying

(f_{w_{l}}f_{w_{l−1}} ⋯ f_{w_{1}}β_{j})(a) = (f_{w_{l}}f_{w_{l−1}} ⋯ f_{w_{1}})(ζ).

By Lemma 1.32 and the Koebe distortion theorem, we can assume that

(f_{w_{l}}f_{w_{l−1}} ⋯ f_{w_{1}}β_{j})(B(a, η)) ⊂ B((f_{w_{l}}f_{w_{l−1}} ⋯ f_{w_{1}})(ζ), ϵ).

Since

(f_{w_{l}}f_{w_{l−1}} ⋯ f_{w_{1}})(z) ∈ B((f_{w_{l}}f_{w_{l−1}} ⋯ f_{w_{1}})(ζ), ϵ),  (2)

(f_{w_{l}}f_{w_{l−1}} ⋯ f_{w_{1}})(y_{j}) = (f_{w_{l}}f_{w_{l−1}} ⋯ f_{w_{1}}β_{j})(α_{n_{j}}(z)) ∈ B((f_{w_{l}}f_{w_{l−1}} ⋯ f_{w_{1}})(ζ), ϵ) (3)

and f_{w_{l+1}}|_{B(f_{w_{l}}f_{w_{l−1}}···f_{w_{1}}(ζ), ϵ)} is injective,

(f_{w_{l+1}}f_{w_{l}} ⋯ f_{w_{1}})(y_{j}) = (f_{w_{l+1}}f_{w_{l}} ⋯ f_{w_{1}})(z)

implies that (f_{w_{l}}f_{w_{l−1}} ⋯ f_{w_{1}})(y_{j}) = (f_{w_{l}}f_{w_{l−1}} ⋯ f_{w_{1}})(z). Hence the claim above holds.

From this claim, it follows that y_{j} = z for each large j. Since diam β_{j}(B(a, η)) → 0 as j → ∞, we have z = ζ.

Theorem 1.34. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated rational semigroup. Assume that F (G) ≠ ∅, there is an element g ∈ G such that deg(g) ≥ 2 and each element of Aut \overline{ℂ}∩G(if this is not empty) is loxodromic. Also we assume all of the following conditions;

1. Ã_{0}(G) is a compact subset of F (G),

2. any element of G with the degree at least two has neither Siegel disks nor Hermann rings.

3. ♯(UH(G) ∩ ∂J(G)) < ∞ and each point of UH(G) ∩ ∂J(G) is not a non-repelling fixed point of any element of G.

Then Ã_{0}(G) = Ã(G) ≠ ∅ and for each compact subset L of F (G),

sup{d(f_{i_{n}} ⋯ f_{i_{1}}(z), Ã(G)) | z ∈ L, (i_{n}, …, i_{1}) ∈ {1, …, m}^{n}} → 0,

as n → ∞, where we denote by d the spherical metric. Also Ã(G) is the smallest attractor in F (G) for G. Moreover we have that if (h_{n}) is a sequence in G consisting of mutually disjoint elements and converges to a map ϕ in a subdomain V of F (G), then ϕ is constant taking its value in Ã(G).

Proof. First we will show that Ã_{0}(G) = Ã(G) ≠ ∅. By the condition 2, g has neither Siegel disks nor Hermann rings. Since F (G) ≠ ∅ and by the condition 3, applying the no wandering domain theorem for ⟨g⟩, we see that the element g has an attracting periodic point x in F (G). Hence Ã_{0}(G) ≠ ∅. By Lemma 1.28, we get Ã_{0}(G) = Ã(G) ≠ ∅.

Next we will show that for each x ∈ F (G), \overline{G(x)} ⊂ F (G). Assume that there exists a connected component U of F (G), a sequence (g_{n}) of elements of G and a point y ∈ ∂J(G) such that (g_{n}) converges to y locally uniformly on U. We take a subsequence (g_{1,n}) of (g_{n}) satisfying that there exists a generator f_{i_{1}} so that

g_{1,n} = ⋯ f_{i_{1}},

for each n. Inductively when we get a sequence (g_{j,n})_{n} satisfying that there exists a word (i_{1}, …, i_{j}) ∈ {1, …, m}^{j} so that g_{j,n} = ⋯ f_{i_{j}} ⋯ f_{i_{1}} for each n, we take a subsequence (g_{j+1,n})_{n} of (g_{j,n})_{n} satisfying that there exists a generator f_{i_{j+1}} so that

g_{j+1,n} = ⋯ f_{i_{j+1}} ⋯ f_{i_{1}}

for each n. By the diagonal method, we get a subsequence (g_{n,n})_{n} of (g_{n}) satisfying that there exists a word (i_{1}, i_{2}, …) ∈ {1, …, m}^{ℕ} so that for each n,

g_{n,n} = α_{n}f_{i_{n}} ⋯ f_{i_{1}},

where α_{n} is an element of G. We consider the sequence (β_{n}) where β_{n} = f_{i_{n}} ⋯ f_{i_{1}}. We see that U_{β_{n}} ≠ U_{β_{m}} if n ≠ m. For, if there exists n and m with n > m such that U_{β_{n}} = U_{β_{m}}, then

(f_{i_{n}} ⋯ f_{i_{m+1}})(U_{β_{m}}) ⊂ U_{β_{m}}

and the element f_{i_{n}} ⋯ f_{i_{m+1}} has an (super)attracting fixed point x_{0} in \overline{U_{β_{m}}}. By the condition 3, we have x_{0} ∈ Ã(G). From Lemma 1.30, it contradicts to that (g_{n}) converges to y ∈ ∂J(G) in U. Hence U_{β_{n}} ≠ U_{β_{m}} if n ≠ m. Now let z be a point of U. Since U_{β_{n}} ≠ U_{β_{m}} if n ≠ m, we have (β_{n}(z)) accumulates only in ∂J(G). By Theorem 1.24, we can show that (β_{n}(z)) accumulates only in ∂J(G) ∩ UH(G). For each large n, let ζ_{n} be the closest point to β_{i_{n}}(z) in ∂J(G) ∩ UH(G). Since ♯(∂J(G) ∩ UH(G)) < ∞ and there is no super attracting fixed point of any element of G in ∂J(G), there exists an integer n_{0} such that for each integer n with n ≥ n_{0},

(f_{i_{n}} ⋯ f_{i_{n_{0}}+1})′(ζ_{n_{0}}) ≠ 0.

¿From Lemma 1.33, we get a contradiction. Therefore we have for each x ∈ F (G), \overline{G(x)} ⊂ F (G).

Now let x be a point of F (G). We have \overline{G(x)} ⊂ F (G). Let \{U_{1}, …, U_{s}\} be the set of all connected components of F (G) having non-empty intersection with \overline{G(x)}. We set V = ∪_{j=1}^{s}U_{j}. Suppose that x ∈ U_{j}. For each (i_{s+1}, i_{s}, …, i_{1}) ∈ {1, …, m}^{s+1}, there exists an integer t with 1 ≤ t ≤ s such that (f_{i_{s+1}} ⋯ f_{i_{t+1}})(U_{j_{t}}) ⊂ U_{j_{t}}, where U_{j_{t}} is the component of V containing (f_{i_{t}} ⋯ f_{i_{1}})(U_{j}). From our assumption, the element f_{i_{s+1}} ⋯ f_{i_{t+1}} has an attracting fixed point in U_{j_{t}} ∩ Ã(G). Hence, from Lemma 1.30, we have

sup{d(f_{i_{n}} ⋯ f_{i_{1}}(z), Ã(G)) | (i_{n}, …, i_{1}) ∈ {1, …, m}^{n}} → 0,

as n → ∞. Therefore for each compact subset L of F (G), the similar result holds.

Next we will show that Ã(G) is the smallest attractor in F (G) for G. From the argument above, Ã(G) is an attractor in F (G) for G. Let K be any attractor in F (G) for G. It is easy to see that each attracting fixed point of any element of G in F (G) belongs to the set K. It implies that Ã(G) ⊂ K.

Finally assume (h_{n}) is a sequence in G consisting of mutually disjoint elements and converges to a map ϕ in a subdomain V of F (G). Then by Lemma 1.30 we see that ϕ is constant taking its value in Ã(G).

By Theorem 1.34 and Lemma 1.29, we get the next theorem.

Theorem 1.35. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated rational semigroup which is semi-hyperbolic. Assume that there is an element g ∈ G such that deg(g) ≥ 2 and each element of Aut \overline{ℂ} ∩ G(if this is not empty) is loxodromic. If F (G) ≠ ∅, then ∅ ≠ A(G) = A_{0}(G) ⊂ F (G) and for each compact subset L of F (G),

sup{d(f_{i_{n}} ⋯ f_{i_{1}}(z), A(G)) | z ∈ L, (i_{n}, …, i_{1}) ∈ {1, …, m}^{n}} → 0,

as n → ∞, where we denote by d the spherical metric. Also A(G) is the smallest attractor in F (G) for G. Moreover we have that if (h_{n}) is a sequence in G consisting of mutually disjoint elements and converges to a map ϕ in a subdomain V of F (G), then ϕ is constant taking its value in Ã(G).

Theorem 1.36. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated rational semigroup which is sub-hyperbolic. Assume that there is an element g ∈ G such that deg(g) ≥ 2 and each element of Aut \overline{ℂ} ∩ G(if this is not empty) is loxodromic. If F (G) ≠ ∅, then ∅ ≠ Ã(G) = Ã_{0}(G) ⊂ F (G) and for each compact subset L of F (G),

sup{d(f_{i_{n}} ⋯ f_{i_{1}}(z), Ã(G)) | z ∈ L, (i_{n}, …, i_{1}) ∈ {1, …, m}^{n}} → 0,

as n → ∞, where we denote by d the spherical metric. Also Ã(G) is the smallest attractor in F (G) for G. Moreover we have that if (h_{n}) is a sequence in G consisting of mutually disjoint elements and converges to a map ϕ in a subdomain V of F (G), then ϕ is constant taking its value in Ã(G).

Proof. Since Ã_{0}(G) ⊂ P (G) and G is sub-hyperbolic, we have that Ã_{0}(G) is a compact subset of F (G) and ♯(UH(G) ∩ J(G)) < ∞. Now let x be a point of UH(G) ∩ ∂J(G). Assume that there exists an element h ∈ G such that h(x) = x. Since G is sub-hyperbolic, x is neither attracting nor indifferent fixed point of h. Since G is finitely generated, by [HM2], we have that there exists no superattracting fixed point of any element of G in ∂J(G). Hence x is a repelling fixed point of h.

From Theorem 1.34, the statement of our theorem holds.

Proposition 1.37. Let G be a finitely generated rational semigroup which contains an element with the degree at least two. Assume that ♯P (G) < ∞ and P (G) ⊂ J(G). Then J(G) = \overline{ℂ}.

Proof. Suppose F (G) ≠ ∅. Let g ∈ G be an element with the degree at least two. By the assumption of our Proposition, g has a super attracting periodic point in ∂J(G). On the other hand, since G is finitely generated, by [HM2], there exist no super attracting fixed points of any element of G in ∂J(G). This is a contradiction.

Definition 1.38. Let M be a complex manifold. Suppose the map

(z, a) ∈ \overline{ℂ} × M ↦ f_{j,a}(z) ∈ \overline{ℂ}

is holomorphic for each j = 1, …, n . We set G_{a} = ⟨f_{1,a}, ⋯, f_{n,a}⟩. Then we say that {G_{a}}_{a∈M} is a holomorphic family of rational semigroups.

By Theorem 1.34 and Theorem 2.3.4 in [S1], we get the following result.

Corollary 1.39. Let M be a complex manifold. Let {G_{a}}_{a∈M} be a holomorphic family of rational semigroups where G_{a} = ⟨f_{1,a}, ⋯, f_{n,a}⟩. Let b be a point of M. We assume that G_{b} satisfies the assumption in Theorem 1.34. Then the map

a ↦ J(G_{a})

is continuous at the point a = b with respect to the Hausdorff metric.

Corollary 1.40. Let M be a complex manifold. Let {G_{a}}_{a∈M} be a holomorphic family of rational semigroups where G_{a} = ⟨f_{1,a}, ⋯, f_{n,a}⟩. Let b be a point of M. Assume that G_{b} contains an element of degree at least two and that each element of Aut \overline{ℂ} ∩ G_{b}(if this is not empty) is loxodromic. If G_{b} is semi-hyperbolic or sub-hyperbolic, then the map

a ↦ J(G_{a})

is continuous at the point a = b with respect to the Hausdorff metric.

2 Rational Skew Product

Definition 2.1 (rational skew product). Let X be a topological space. If a continuous map f̃ : X × \overline{ℂ} → X × \overline{ℂ} is represented by the following form:

f̃((x, y)) = (p(x), q_{x}(y)),

where p : X → X is a continuous map and q_{x} : \overline{ℂ} → \overline{ℂ} is a rational map with the degree at least 1 for each x ∈ X, then we say that f̃: X × \overline{ℂ} → X × \overline{ℂ} is a rational skew product. In this paper we always assume that X is a compact metric space.

O.Sester investigated polynomial skew products(in particular, quadratic case) in [Se]. M.Jonsson investigated dynamics on \overline{ℂ}-fibration whose fiberwise maps are rational maps of degree d, d ≥ 2 in [J2].

Definition 2.2. Let G be a rational semigroup generated by {f_{λ}}_{λ∈Λ}. Let X = Λ^{ℕ}. Let f̃: X × \overline{ℂ} → X × \overline{ℂ} be the map defined by:

f̃((x, y)) = (p(x), f_{x_{1}}(y)),

where p : X → X is the shift map and x ∈ X is represented by: x = (x_{1}, x_{2}, …). Then we say that f̃ : X × \overline{ℂ} → X × \overline{ℂ} is the rational skew product constructed by the generator system {f_{λ}}_{λ∈Λ}.

Definition 2.3. Let f̃: X × \overline{ℂ} → X × \overline{ℂ} be a rational skew product. For each n ∈ ℕ and x ∈ X, we set q_{x}^{(n)} := q_{p^{n−1}(x)}∘⋯∘q_{x} and f̃_{x}^{n} := f̃^{n}|_{π_{X}^{−1}({x})}. We define the following sets. For each x ∈ X,

d(x) = deg(q_{x}),
F_{x} = \{y ∈ \overline{ℂ} | \{q_{x}^{(n)}\}_{n} is normal in a neigborhood of y\},
J_{x} = \overline{ℂ} \ F_{x}, J̃_{x} = \{x\} × J_{x}.

Further we set

J̃(f̃) = \overline{⋃_{x∈X} J̃_{x}}, F̃(f̃) = (X × \overline{ℂ}) \ J̃(f̃).

C(f̃) = \{(x, y) ∈ X × \overline{ℂ} | q′_{x}(y) = 0\}, P (f̃) = \overline{⋃_{n∈ℕ} f̃^{n}(C(f̃))}.

C(f̃) is called the critical set for f̃ and P (f̃) is called the post critical set for f̃. Moreover we set

(f̃^{n})′((x, y)) = (q_{x}^{(n)})′(y).

If (x, y) is a period point of f̃ with the period n, then we say that (x, y) is repelling(resp. indifferent, attracting, etc.) if |(f̃^{n})′((x, y))| > 1(resp.= 1, < 1, etc.).

Lemma 2.4. Let f̃: X ×\overline{ℂ} → X ×\overline{ℂ} be a rational skew product represented by f̃((x, y)) = (p(x), q_{x}(y)). Then the following hold.

1. if x ∈ X, then q_{x}^{−1}(F_{p(x)}) = F_{x}, q_{x}^{−1}(J_{p(x)}) = J_{x}, f̃(J̃(f̃)) ⊂ J̃(f̃).

2. if p : X → X is surjective, then f̃: X × \overline{ℂ} → X × \overline{ℂ} is surjective.

3. if p : X → X is a surjective and open map, then f̃^{−1}(J̃(f̃)) = f̃(J̃(f̃)) = J̃(f̃).

Proof. We will show the last statement. Suppose f̃((x, y)) ∈ J̃(f̃). Then there exists a sequence ((x_{i}, y_{i})) converging to f̃((x, y)) such that y_{i} ∈ F_{x_{i}} for each i. Since p is an open map, there exists a sequence (x̃_{i}) converging to x such that p(x̃_{i}) = x_{i}. Then there exists a sequence (ỹ_{i}) converging to y such that qx̃_{i}(ỹ_{i}) = y_{i} for each i. Then ỹ_{i} ∈ Jx̃_{i}. Hence (x, y) ∈ J̃(f̃). Hence f̃^{−1}(J̃(f̃)) ⊂ J̃(f̃). Since f̃(J̃)(f̃) ⊂ J̃(f̃) and f̃: X × \overline{ℂ} → X × \overline{ℂ} is surjective, we have f̃^{−1}(J̃(f̃)) = f̃(J̃(f̃)) = J̃(f̃).

Now we need some notations from [J2], concerning potential theoritic aspects. Let f̃: X × \overline{ℂ} → X × \overline{ℂ} be a rational skew product represented by f̃((x, y)) = (p(x), q_{x}(y)). Let ω be the spherical probability measure on \overline{ℂ}. Let ω_{x} = (i_{x})_{∗}ω for each x ∈ X where we denote by i_{x} : \overline{ℂ} → π_{X}^{−1}(\{x\}) the natural isomorphism. For each continuous function φ on π_{X}^{−1}({x}) let (f̃_{x}^{n})^{∗}φ be the continuous function on π_{X}^{−1}({p^{n}(x)}) defined by ((f̃_{x}^{n})^{∗}φ)(z) = ∑_{f̃_{x}^{n}(w)=z} φ(w) for each n ∈ ℕ. Let μ_{x,n} be the probability measure on π_{X}^{−1}({x})

defined by ⟨μ_{x,n}, φ⟩ = _{\frac{1}{∏_{j=0}^{n−1} d(p^{j}(x))}}⟨ω_{p^{n}(x)}, (f̃_{x}^{n})^{∗}φ⟩. For each x ∈ X, we

denote by R_{x} : ℂ^{2} \ {0} → ℂ^{2} \ {0} the homogenious polynomial mapping of degree d(x) such that q_{x} ∘π′ = π′ ∘R_{x} where π′ : ℂ^{2} \\{0\} → \overline{ℂ} is the natural projection and sup{|R_{x}(z, w)| | |(z, w)| = 1} = 1. R_{x} is determined uniquely up to multiplication by a complex number of units. We can assume x ↦ R_{x} is continuous. For each x ∈ X and n ∈ ℕ let G_{x,n} := _{\frac{1}{∏_{j=0}^{n−1} d(p^{j}(x))}} log |R_{x}^{n}|

where R_{x}^{n} := R_{p^{n−1}(x)} ∘ ⋯ ∘ R_{x}. Then the following results hold.

Proposition 2.5. Let f̃ : X × \overline{ℂ} → X × \overline{ℂ} be a rational skew product represented by f̃((x, y)) = (p(x), q_{x}(y)) and assume d(x) ≥ 2 for each x ∈ X. Then we have the following.

1. μ_{x,n} converges to a probability measure μ_{x} on π_{X}^{−1}({x}) weakly as n → ∞ for each x ∈ X.

2. G_{x,n} converges to a continuous plurisubharmonic function G_{x} locally uniformly on ℂ^{2} \ {0} as n → ∞ for each x ∈ X.

3. μ_{x} = (i_{x}^{−1})_{∗}(dd^{c}(G_{x} ∘s)) where s is a local section of π′ : ℂ^{2} \\{0\} → \overline{ℂ}. Further G_{x}(z, w) ≤ log |(z, w)|+O(1) as |(z, w)| → ∞ and G_{x}(λz, λw) = G_{x}(z, w) + log λ for each λ ∈ ℂ, for each x ∈ X.

4. G_{p(x)} ∘ R_{x} = d(x) · G_{x} for each x ∈ X.

5. if x → x′ then G_{x} → G_{x′} uniformly on ℂ^{2} \ {0}.

6. (f̃_{x})_{∗}μ_{x} = μ_{p(x)}, (f̃_{x})^{∗}μ_{p(x)} = d(p(x)) · μ_{x} for each x ∈ X.

7. μ_{x} puts no mass on polar subsets of π_{X}^{−1}({x}) for each x ∈ X.

8. x ↦ μ_{x} is continuous with respct to the weak topology of measures in X × \overline{ℂ}.

9. supp(μ_{x}) = J̃_{x} for each x ∈ X.

10. J̃_{x} has no isolated points for each x ∈ X.

11. x ↦ J̃_{x} is lower semicontinuous with respect to the Hausdorff metric in the space of compact subsets of X × \overline{ℂ}.

Proof. Since d(x) ≥ 2 for each x ∈ X, we can show the statements in the same way as that in section 3 in [J2].

Definition 2.6 (hyperbolicity). Let f̃ : X × \overline{ℂ} → X × \overline{ℂ} be a rational skew product. We say that f̃ is hyperbolic along fibres if P (f̃) ⊂ F̃(f̃).

Definition 2.7. Let f̃: X × \overline{ℂ} → X × \overline{ℂ} be a rational skew product. We say that f̃ is expanding along fibres if there exists a positive constant C and a constant λ with λ > 1 such that for each n ∈ ℕ,

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

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

Definition 2.8 (semi-hyperbolicity). Let f̃ : X × \overline{ℂ} → X × \overline{ℂ} be a rational skew product. Let N be a positive integer. We say that a point (x_{0}, y_{0}) ∈ X × \overline{ℂ} belongs to SH_{N}(f̃) if there exists a neighborhood U of x_{0} and a positive number δ satisfying that for any x ∈ U, any n ∈ ℕ, any element x_{n} ∈ p^{−n}(x) and any element V of c(B(y_{0}, δ), q_{x_{n}}^{(n)}),

deg(q_{x_{n}}^{(n)} : V → B(y_{0}, δ)) ≤ N.

We set

UH(f̃) = (X × \overline{ℂ}) \ ∪_{N∈ℕ}SH_{N}(f̃).

We say that f̃ is semi-hyperbolic along fibres if for any (x_{0}, y_{0}) ∈ J̃(f̃) there exists a positive integer N such that (x_{0}, y_{0}) ∈ SH_{N}(f̃).

Lemma 2.9. Let f̃: X × \overline{ℂ} → X × \overline{ℂ} be a rational skew product. If f̃ is hyperbolic along fibres, then it is semi-hyperbolic along fibres.

Lemma 2.10. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated rational semigroup. Then G is semi-hyperbolic if and only if the rational skew product f̃: X ×\overline{ℂ} → X ×\overline{ℂ} constructed by the generator system \{f_{1}, f_{2}, …, f_{m}\} is semi-hyperbolic along fibres. G is hyperbolic if and only if f̃ is hyperbolic along fibres.

Definition 2.11 (Condition(C1)). Let f̃: X × \overline{ℂ} → X × \overline{ℂ} be a rational skew product. We say that f̃ satisfies the condition (C1) if there exists a family \{D_{x}\}_{x∈X} of discs in \overline{ℂ} such that the following three conditions are satisfied:

1. \overline{⋃_{n≥0} f̃^{n}(\{x\} × D_{x})} ⊂ F̃(f̃).

2. for any x ∈ X, we have that diam(q_{x}^{(n)}(D_{x})) → 0, as n → ∞.

3. inf_{x∈X} diam (D_{x}) > 0.

Now we will show the following lemma and theorem.

Lemma 2.12. Let f̃: X × \overline{ℂ} → X × \overline{ℂ} be a rational skew product satisfying the condition (C1). Assume that there exists a point (x_{0}, y_{0}) ∈ X × \overline{ℂ} with y_{0} ∈ F_{x_{0}}, a connected open neighborhood U of y_{0} in \overline{ℂ} and a sequence (n_{j}) of positive integers such that R_{j} := q_{x_{0}}^{(n_{j})} converges to a non-constant map ϕ uniformly on U as j → ∞. Let (x_{j}, y_{j}) = f̃^{n_{j}}(x_{0}, y_{0}) and (x_{∞}, y_{∞}) = lim_{j→∞}(x_{j}, y_{j}). Let S_{i,j} = q_{x_{i}}^{(n_{j}−n_{i})} for 1 ≤ i < j. Let

V = \{y ∈ \overline{ℂ} | ∃ϵ > 0, lim_{i→∞} sup_{j>i} sup_{d(ξ,y)≤ϵ} d(S_{i,j}(ξ), ξ) = 0\}.

Then V is a non-empty open set and for any y ∈ ∂V, we have that

(x_{∞}, y) ∈ J̃(f̃) ∩ UH(f̃).  (4)

Theorem 2.13. Let f̃: X ×\overline{ℂ} → X ×\overline{ℂ} be a rational skew product. Assume f̃ is semi-hyperbolic along fibres and satisfies the condition (C1). Then the following hold.

1. Let (x_{0}, y_{0}) ∈ X × \overline{ℂ} be any point with y_{0} ∈ F_{x_{0}}. Then for any open connected neighborhood U of y_{0} in \overline{ℂ}, there exists no subsequence of (q_{x_{0}}^{(n)})_{n} converging to a non-constant map locally uniformly on U.

2.

J̃(f̃) = ⋃_{x∈X} J̃_{x}.

3. If there exists a disc D in \overline{ℂ} such that D_{x} = D for all x ∈ X in the condition (C1), then there exist positive constants δ, L and λ(0 < λ < 1) such that for any n ∈ ℕ,

sup{diam U | U ∈ c(B(y, δ), q_{x_{n}}^{(n)}), (x, y) ∈ J̃(f̃), x_{n} ∈ p^{−n}(x)} ≤ Lλ^{n}.

4. Assume d(x) ≥ 2 for each x ∈ X. Then we have that x ↦ J̃_{x} is continuous with respect to the Hausdorff metric in the space of compact subsets of X × \overline{ℂ}.

5. Assume d(x) ≥ 2 for each x ∈ X. Then for any compact subset K of F̃(f̃), we have that \overline{∪_{n≥0}f̃^{n}(K)} ⊂ F̃(f̃) and there exist constants C > 0 and τ < 1 such that for each n, sup_{z∈K} ∥(f̃^{n})′(z)∥ ≤ Cτ^{n}.

To show Lemma 2.12 and Theorem 2.13, we need the following lemma.

Lemma 2.14. Let f̃: X × \overline{ℂ} → X × \overline{ℂ} be a rational skew product satisfying the condition (C1). Assume (x_{0}, y_{0}) ∈ SH_{N}(f̃) 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} in \overline{ℂ} satisfying that for each n ∈ ℕ, each x ∈ U and each x_{n} ∈ p^{−n}(x), we have that each element of c(B(y_{0}, δ), q_{x_{n}}^{(n)}) is simply connected.

Proof. Take a positive number δ_{1} such that for each x ∈ X and each x_{1} ∈ p^{−1}(x), we have that each connected component of q_{x_{1}}^{−1}(D_{x}) contains a ball with the radius at least δ_{1}.

By the semi-hyperbolicity and Lemma 1.10, we can take a positive number δ_{0} and a open neighborhood U of x_{0} in \overline{ℂ} such that for each δ with 0 < δ < δ_{0} , each x ∈ U, each n ∈ ℕ, and each x_{n} ∈ p^{−n}(x), we have that the diameter of each element of c(B(y_{0}, δ), q_{x_{n}}^{(n)}) is less than δ_{1}.

Now we will show each element of c(B(y_{0}, δ), q_{x_{n}}^{(n)}) is simply connected by induction on n. Assume an element W of c(B(y_{0}, δ), q_{x_{n}}^{(n)}) is simply connected. Let W_{1} be a connected component of q_{x_{n+1}}^{−1}(W) where x_{n+1} is an element of p^{−1}(x_{n}). Suppose W_{1} is not simply connected. Each connected component of ∂W_{1} is mapped onto ∂W by q_{x_{n+1}}. Hence the image of each connected component of \overline{ℂ} \ W_{1} by q_{x_{n+1}} contains D_{x_{n}}. Hence we have that diam W_{1} ≥ δ_{1} , which contradicts to the choice of δ_{0} and U.

Now we will show the Lemma 2.12.

Proof. We will show the statement developing a method in M.Jonsson’s Thesis([J1]). By the definition, V is an open set. Since ϕ is non-constant, there exists a positive number a such that

R_{j}(U) ⊃ B(y_{∞}, a)

for each j ∈ ℕ. We have that B(y_{∞}, a) ⊂ V. For, if y ∈ B(y_{∞}, a) then y = R_{i}(ξ_{i}) for some ξ_{i} ∈ U and so d(S_{i,j}(y), y) = d(R_{j}(ξ_{i}), R_{i}(ξ_{i})) which is small if i is large. Hence V is a non-empty open set.

Take any y ∈ ∂V. We will show

(x_{∞}, y) ∈ J̃(f̃).  (5)

Assume this is false. If there exists a positive integer i_{0} such that {S_{i,j}}_{j≥i≥i_{0}} is normal in a neighborhood of y, then since S_{i,j} → Id on V ∩ W, we have that W ⊂ V and it is a contradiction. Hence there exist sequences (i_{k}), (j_{k}) and (ξ_{k}) such that i_{k} ≤ j_{k}, j_{k} − i_{k} → ∞, ξ_{k} → y and

S_{i_{k}, j_{k}}(ξ_{k}) ∈ D_{x_{i_{k}}} ,  (6)

where we denote by (D_{x})_{x∈X} a family of discs in \overline{ℂ} in the definition of condition (C1). Since we are assuming (x_{∞}, y) ∈ F̃(f̃), we have that there exists an disc B around y such that B ⊂ F_{x_{i_{k}}} and ξ_{k} ∈ B for large k. By (6), the condition (C1) and the definition of V, we get a contradiction. Hence (5) holds.

Now we will show (x_{∞}, y) ∈ UH(f̃). Suppose this is false. Then there exists a positive integer N such that (x_{∞}, y) ∈ SH_{N}(f̃). Let δ_{0} be a number for (x_{∞}, y) in Lemma 2.14 and let δ = δ_{0}/2. We can assume that there exists a neighborhood U′ of x_{∞} satisfying that for any x ∈ U′, any n ∈ ℕ, any element x_{n} ∈ p^{−n}(x) and any element V of c(B(y, δ_{0}), q_{x_{n}}^{(n)}),

deg(q_{x_{n}}^{(n)} : V → B(y, δ_{0})) ≤ N.

Take two domains V_{1} and V_{2} such that

y_{∞} ∈ V_{2} ⊂⊂ V_{1} ⊂⊂ V, B(y, δ) ∩ V_{2} ≠ ∅.  (7)

If j > i and i is large enough, then S_{i,j} is close to id_{V_{1}} on V_{1}. Hence S_{i,j} is biholomorphic on V_{1} and S_{i,j}(V_{1}) ⊃ V_{2}. Let h_{i,j} : V_{2} → V_{1} be a map such that S_{i,j} ∘ h_{i,j} = id on V_{2} and h_{i,j} ∘ S_{i,j} = id on S_{i,j}^{−1}(V_{2}) ∩ V_{1}. Then we have that

lim_{i→∞} sup_{j>i} sup_{ξ∈V_{2}} d(h_{i,j}(ξ), ξ) = 0.  (8)

For each (i, j) such that j > i and i is large enough, let B_{i,j} ∈ c(B(y, δ), S_{i,j}) be an element such that h_{i,j}(V_{2}∩B(y, δ)) ⊂ B_{i,j}. By the choice of δ_{0}, we have that B_{i,j} is simply connected. By semi-hyperbolicity, there exists a positive integer M such that for each (i, j) with j > i where i is large enough,

♯( cv(S_{i,j}|_{B_{i,j}}) ∩ B(y, δ)) ≤ M,  (9)

where we denote by “cv” the set of critical values. Hence there exists a positive number θ with 0 < θ < 2π such that for each (i, j) with i < j there exists a sector U_{i,j} in B(y, δ) of angle θ with the center y such that U_{i,j} ∩ cv(S_{i,j}|_{B_{i,j}}) = ∅ and U_{i,j} ∩ V_{2} ≠ ∅. Let g_{i,j} : U_{i,j} → B_{i,j} be the analytic continuation of h_{i,j} on V_{2} ∩ U_{i,j}. Let y_{i,j} ∈ B_{i,j} ∩ \overline{g_{i,j}(U_{i,j})} such that S_{i,j}(y_{i,j})) = y. By (8), Corollary 1.9, Condition (C1) and the fact V_{2} ∩ U_{i,j} ≠ ∅, we have that there exists a positive number δ_{1} such that

B(y_{i,j}, δ_{1}) ⊂ B_{i,j},  (10)

for each (i, j) such that j > i and i is large enough. Now we will show the following claim:

Claim: lim_{i→∞} sup_{j>i} d(y_{i,j}, y) = 0.  (11)

Suppose this is false. Then there exists a sequence ((i_{k}, j_{k})) with j_{k} > i_{k} and a positive number δ_{2} such that d(y_{i_{k},j_{k}}, y) > δ_{2}, for each k. We can assume that (y_{i_{k},j_{k}}) converges to a point ỹ as k → ∞ and that there exists a sector U_{0} with the center y such that U_{i_{k},j_{k}} = U_{0} for each k. Then we have that

B(ỹ, δ_{1}/2) ∩ V = ∅.  (12)

For, assume the left hand side is not empty. Then since S_{i_{k},j_{k}} → id inV and the family {S_{i_{k},j_{k}}|_{B(ỹ,δ_{1}/2)}}_{k} is normal, we have that S_{i_{k},j_{k}} → id locally uniformly on B(ỹ, δ_{1}/2). But this is a contradiction because ỹ ≠ y. Hence we have (12).

By Lemma 1.10, there exists a positive number δ_{3} with δ_{3} < δ_{1}/4 such that for each k, the diameter of each element of c(B(y, δ_{3}), S_{i_{k},j_{k}}) is less than δ_{1}. Hence if we take a fixed point z ∈ B(y, δ_{3}) ∩ U_{0}, then we have that for each large k,

d(ỹ, g_{i_{k},j_{k}}(z)) < δ_{1}/4.  (13)

On the other hand, since g_{i_{k},j_{k}} → id locally uniformly on V_{2} and (g_{i_{k},j_{k}})_{k} is normal in U_{0}, we have that g_{i_{k},j_{k}} → id locally uniformly on U_{0}. Hence we have that d(g_{i_{k},j_{k}}(z), y) < δ_{1}/4 for each large k. Together with (13) and B(ỹ, δ_{1}/2)∩V = ∅, we get a contradiction. Hence we have shown the claim (11).

Since B(y_{i,j}, δ_{1}) ⊂ B_{i,j} for each (i, j) such that j > i and i is large enough, by the above claim we have that there exists a positive integer i_{0} such that for each (i, j) with j > i ≥ i_{0},

S_{i,j}(B(y, δ_{1}/2)) ⊂ B(y, δ).

Hence (S_{i,j})_{j>i≤i_{0}} is normal in B(y, δ_{1}/2). Since S_{i,j} → id on B(y, δ_{1}/2)∩V, we have that y ∈ V and this is a contradiction. Hence we have shown the first statement of our lemma.

Now we will show Theorem 2.13.

Proof. The statement 1 follows from Lemma 2.12.

Now we will show the statement 2 of our theorem. Suppose the statement is false. Then there exists a point (x_{0}, y_{0}) ∈ J̃(f̃) with y_{0} ∈ F_{x_{0}}, a connected component U of y_{0} in \overline{ℂ} and a sequence (n_{j}) of positive integers such that R_{j} := q_{x_{0}}^{(n_{j})} converges to a map ϕ uniformly on U as j → ∞. Let (x_{j}, y_{j}) = f̃^{n_{j}}(x_{0}, y_{0}) and (x_{∞}, y_{∞}) = lim_{j→∞}(x_{j}, y_{j}). By Lemma 1.10, there exists a positive number a such that y_{j} ∈ B(y_{∞}, a) and the element B_{j} ∈ c(B(y_{∞}, a), R_{j}) containing y_{j} satisfies that B_{j} ⊂ U for each large j. Hence R_{j}(U) ⊃ B(y_{∞}, a) for each large j and it implies that ϕ is nonconstant. By Lemma 2.12, it is a contradiction. Hence we have shown the statement 2 of our theorem.

Next we will show the statement 3 of our theorem. By semi-hyperbolicity and Lemma 2.14, there exists a positive integer N and a positive real number δ_{0} such that for any (x′, y′) ∈ J̃(f̃) there exists a neighborhood U′ of x′ satisfying that for any real number τ with 0 < τ ≤ δ_{0}, any x ∈ U′, any n ∈ ℕ, any element x_{n} ∈ p^{−n}(x) and any element V of c(B(y′, τ), q_{x_{n}}^{(n)}), we have that V is simply connected and

deg(q_{x_{n}}^{(n)} : V → B(y′, τ)) ≤ N.

Let δ = δ_{0}/2. We set

A_{n} = sup{diam U | U ∈ c(B(y, δ), q_{x_{n}}^{(n)}), (x, y) ∈ J̃(f̃), x_{n} ∈ p^{−n}(x)}.

First we will show

A_{n} → 0, as n → ∞.  (14)

Suppose this is false. Then there exists a positive constant C, a sequence ((x^{k}, y^{k})) of points in J̃(f̃), a sequence ((x̃^{k}, ỹ^{k})) with f̃^{n_{k}}((x̃^{k}, ỹ^{k})) = (x^{k}, y^{k}) for some n_{k} ∈ ℕ, → ∞ and a sequence (U_{k})_{k} with U_{k} ∈ c(B(y_{k}, δ), q_{x̃^{k}}^{(n_{k})}) and ỹ^{k} ∈ U_{k} for each k such that

diam U_{k} ≥ C, for each k.

We can assume that ((x^{k}, y^{k})) tends to a point (x^{0}, y^{0}) ∈ J̃(f̃) and that ((x̃^{k}, ỹ^{k})) tends to a point (x̃^{0}, ỹ^{0}) ∈ J̃(f̃). By Corollary 1.9, there exists a positive number r such that B(ỹ^{0}, r) ⊂ U_{k} for each large k. Hence

q_{x̃^{k}}^{(n_{k})}(B(ỹ^{0}, r)) ⊂ B(y^{0}, r),  (15)

for each large k. By the second statement of our theorem, we have that ỹ^{0} ∈ J_{}_{x̃^{0}}. Hence there exists a positive integer j and a point z ∈ B(ỹ^{0}, r) such that

q_{x̃^{0}}^{(j)}(z) ∈ D.

Hence q_{x̃^{k}}^{(j)}(z) ∈ D for each large k. On the other hand by the condition (C1) if we take δ_{0} so small then we can assume

\overline{⋃_{n≥0} f̃^{n}(\{x\} × D)} ∩ (\{y^{0}\} × B(y^{0}, δ_{0})) = ∅.

Since n_{k} → ∞, by (15) we get a contradiction. Hence we have (14).

Take a positive integer n_{0} such that for each n ∈ ℕ with n ≥ n_{0},

A_{n} ≤ δ/2.  (16)

Fix any positive integer k. Let ((x_{n}, y_{n})) be a sequence such that f̃((x_{n+1}, y_{n+1})) = (x_{n}, y_{n}) for each n and (x_{0}, y_{0}) ∈ J̃(f̃). For each j = 0, …, k, let W_{j} be the element of c(B(y_{(k−j)n_{0}}, δ), q_{x_{kn_{0}}}^{(jn_{0})}) containing y_{kn_{0}}. By (16) we have that

W_{0} ⊃ ⋯ ⊃ W_{k}.  (17)

For each j = 1, …, k,

q_{x_{kn_{0}}}^{(jn_{0})} : W_{j} → B(y_{(k−j)n_{0}}, δ)

is a proper holomorphic map with the degree at most N. Since q_{x_{kn_{0}}}^{(jn_{0})}(W_{j+1}) is a connected component of (q_{x_{(k−j)n_{0}}}^{(n_{0})} )^{−1}(B(y_{(k−j−1)n_{0}}, δ)), which is included in B(y_{(k−j)n_{0}}, δ/2) by (16), we have that for each j = 0, …, k − 1,

mod (\overline{W_{j+1}}, W_{j}) ≥ c > 0,  (18)

where c is a constant number depending only on N. By Lemma 1.11, there exists a λ with 0 < λ < 1 depending only on N such that

diam W_{j+1}/diam W_{j} ≤ λ, for each j = 0, …, k − 1.

Hence we get that diam W_{k} ≤ λ^{k} diam B(y_{0}, δ). Therefore the statement 3 of our theorem holds.

The statement 4 of our theorem follows from the statement 2 of Theorem 2.13 and 11 in Proposition 2.5.

Next we will show the statement 5 of our theorem. Let K be a compact subset of F̃(f̃). Suppose there exists a sequence (x_{k}, y_{k}) of points in K and a sequence (n_{k}) of positive integers such that (f̃^{n_{k}}((x_{k}, y_{k}))) converges to a point (x′, y′) ∈ J̃(f̃). By 11 in Proposition 2.5, we have that x ↦ J̃_{x} is lower semi-continuous. Hence we have that for each k there exists a point z_{k} ∈ J_{p^{n_{k}} (x_{k})} and (z_{k}) converges to y′. Since K ∩J̃(f̃) = ∅, by the condition (C1)

and Lemma 1.10 we get a contradiction. Hence we get that \overline{∪_{n≥0}f̃^{n}(K)} ⊂ F̃(f̃). Let K′ = \overline{∪_{n≥0}f̃^{n}(K)}. By the statement 1 of Theorem 2.13, we have that for each z = (x, y) ∈ K′, there exists a neighoborhoood U(z) of x in X, a neighborhood V (z) of y in \overline{ℂ} and a positive integer n(z) such that ∥(f̃^{(n(z))})′(z′)∥ < 1/2 for each z′ ∈ U(z) × V (z). Since K′ is a compact set, we get the statement .

Corollary 2.15. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated rational semigroup which is semi-hyperbolic. Assume G contains an element with the degree at least two and each element of Aut \overline{ℂ} ∩ G(if this is not empty) is loxodromic. Also assume F (G) ≠ ∅. Then there exists a δ > 0, a constant L with L > 0 and a constant λ with 0 < λ < 1 such that

sup{diam U | U ∈ c(B(x, δ), f_{i_{n}} ⋯ f_{i_{1}}), x ∈ J(G), (i_{1}, …, i_{n}) ∈ {1, …, m}^{n}}

≤ Lλ^{n}, for each n.

Proof. Let f̃ : X × \overline{ℂ} → X × \overline{ℂ} be the rational skew product constructed by the generator system {f_{1}, f_{2}, …, f_{m}}. Then this is semi-hyperbolic along fibres. By the existence of an attractor in F (G) for G (Theorem 1.35) we have that if we set D_{x} = D for each x ∈ X where D is a small disc around a point of the attractor, then f̃ satisfies the condition (C1) with that family of discs. By Theorem 2.13, the statement of our Corollary holds.

Theorem 2.16. Let f̃: X ×\overline{ℂ} → X ×\overline{ℂ} be a rational skew product. Assume f̃ is hyperbolic along fibres and satisfies the condition (C1) with a family of discs (D_{x})_{x∈X} such that there exists a disc D satisfying D_{x} = D for all x ∈ X. Then f̃ is expanding along fibres.

Proof. We have only to show that there exists a positive integer n_{0} such that for each n ∈ ℕ with n ≥ n_{0} and z ∈ J̃(f̃),

∥(f̃^{n})′(z)∥ ≥ 2.

Suppose this is false. Then there exists a sequence (n_{j}) of positive integers and a sequence (z_{j}) = ((x_{j}, y_{j})) in J̃(f̃) such that

∥(f̃^{n_{j}})′(z_{j})∥ ≤ 2.  (19)

We can assume that f̃^{n_{j}}(z_{j}) converges to a point (x, y) ∈ J̃(f̃). Let δ be a small positive number. For each j let B_{j} ∈ c(B(y, δ), q_{x_{j}}^{(n_{j})}) be the element containing y_{j}. By (19) and Koebe distortion theorem, there exists a positive constant c such that for each j, diam B_{j} ≥ c. But this contradicts to the statement 3 of Theorem 2.13.

Remark 5. We can show that the results in this section are generalized to the version of dynamics on \overline{ℂ}-fibration. For the definition of \overline{ℂ}-fibration, see [J2].

3 Conditions to be semi-hyperbolic

Theorem 3.1. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated rational semigroup. Let z_{0} ∈ J(G) be a point. Assume all of the following conditions:

1. there exists a neighborhood U_{1} of z_{0} in \overline{ℂ} such that for any sequence (g_{n}) ⊂ G, any domain V in \overline{ℂ} and any point ζ ∈ U_{1}, we have that the sequence (g_{n}) does NOT converge to ζ locally uniformly on V.

2. there exists a neighborhood U_{2} of z_{0} in \overline{ℂ} and a positive real number ϵ̃ such that if we set

T = \{c ∈ \overline{ℂ} | ∃j, f′_{j}(c) = 0, (G ∪ \{id\})(f_{j}(c)) ∩ U_{2} ≠ ∅\}

then for each c ∈ T ∩ C(f_{j}), we have d(c, (G ∪ {id})(f_{j}(c))) > ϵ̃.

3. F (G) ≠ ∅.

Then z_{0} ∈ SH_{N}(G) for some N ∈ ℕ.

Notation: For any family {g_{λ}}_{λ∈Λ} of rational functions, we denote by F (\{g_{λ}\}) the set of all points z ∈ \overline{ℂ} such that z has a neighborhood where the family \{g_{λ}\} is normal. We set J(\{g_{λ}\}) = \overline{ℂ} \ F (\{g_{λ}\}). F (\{g_{λ}\}) is called the Fatou set and J({g_{λ}}) is called the Julia set for the family.

Corollary 3.2. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated rational semigroup. Let z_{0} ∈ J(G) be a point. Assume all of the following conditions:

1. there exists a neighborhood U_{1} of z_{0} in \overline{ℂ} such that for any sequence (g_{n}) ⊂ G consisting of mutually distinct elements and any domain V in F ((g_{n})), there exists a point x ∈ V such that the sequence \overline{∪_{n}\{g_{n}(x)\}} ∩ \overline{ℂ} \ U_{1} ≠ ∅.

2. there exists a neighborhood U_{2} of z_{0} in \overline{ℂ} and a positive real number ϵ̃ such that if we set

T = \{c ∈ \overline{ℂ} | ∃j, f′_{j}(c) = 0, (G ∪ \{id\})(f_{j}(c)) ∩ U_{2} ≠ ∅\}

then for each c ∈ T ∩ C(f_{j}), we have d(c, (G ∪ {id})(f_{j}(c))) > ϵ̃.

3. F (G) ≠ ∅.

Then z_{0} ∈ SH_{N}(G) for some N ∈ ℕ and there exists a neighborhood W of z_{0} in \overline{ℂ} such that for any sequence (g_{n}) ⊂ G consisting of mutually distinct elements, we have

sup{ diam S | S ∈ c(W, g_{n})} → 0, as n → ∞.

We will consider the proof of Theorem 3.1. We may assume U_{1} = U_{2} = U for some small disc U . By condition 1 and 3, we may assume ∞ ∈ F (G) and g^{−1}(U) ⊂ ℂ for each g ∈ G. Now we will show the above theorem by developing a lemma in [Ma] and using the methods in [KS]. The stories are almost same as those in [KS], except some modifications.

First we need some new notations. An “square” is a set S of the form

S = {z ∈ ℂ | |ℜ(z − p)| < δ, |ℑ(z − p)| < δ}.

The point p is called the center of S and δ its radius . For each k > 0, given a square S with center p and radius δ, we denote by S^{k} the square with the center p and radius kδ. Take a σ > 0 such that U contains a closed square Q′ with the center a point in U and its radius 2σ. Let Q′′ = (Q′)^{1/2}. Q′′ is called “admissable square at level 1.” We will define asmissable squares at level n for each n ∈ ℕ. Let Q be an admissable square at level n with the radius a. Then Q is covered by 16 squares with the radius a/8. We have 20 squares with the radius a/8 adjacent to Q. We call all these 36 squares admissable at level n+1. These squares are denoted by {Q_{μ,n+1}}. The union of these 36 squares is denoted by Q̃, which is called the “square attached to Q.” Each admissable and each attached square is a relative compact subset of U.

Notation: For any open set V_{1} and for any rational map g, if V_{2} ∈ c(V_{1}, g) then we set △(V_{1}, g) = ♯{x ∈ V_{1} | g′(x) = 0}, counting the multiplicity.

We need some lemmas to show Theorem 3.1.

Lemma 3.3. For given ϵ > 0 and N ∈ ℕ, there exists some n_{0} ∈ ℕ such that the following holds: If Q is an admissable square at some level n ≥ n_{0}, Q̃ the corresponding attached square, V an element of c(Q̃, f) for some f ∈ G, and △(V, f) ≤ N, then diam (K) ≤ ϵ for each element K ∈ c(Q, f) contained in V.

Proof. Fix ϵ > 0 and N ∈ ℕ. If the lemma is false then there exists a sequence (n_{k})_{k∈ℕ} converging to ∞, admissable squares Q_{μ_{k},n_{k}} and functions g_{k} ∈ G such that diam (K_{k}) ≥ ϵ > 0 and △(V_{k}, g_{k}) ≤ N for some element V_{k} ∈ c(Q̃_{μ_{k},n_{k}}, g_{k}) and some element K_{k} ∈ c(Q_{μ_{k},n_{k}}, g_{k}) contained in V_{k}. Take γ > 1 such that γ^{N+1} = \frac{3}{2}. Then there exists 0 < j ≤ N + 1 such that, denoting by R̂_{k} = Q_{μ_{k},n_{k}}^{γ^{j}}, the set R̂_{k}^{γ} − R̂_{k} does not contain any critical values of g_{k}|_{V_{k}}. We have R̂_{k} ⊃ Q_{μ_{k},n_{k}}. Take K̂_{k} ∈ c(R̂_{k}, g_{k}) such that K_{k} ⊂ K̂_{k} ⊂ V_{k}. Then diam K̂_{k} > ϵ. The element g_{k} is represented by the following form: g_{k} = f_{s_{l}} ∘ ⋯ ∘ f_{s_{1}}. Then there exists a positive integer i such that

diam f_{s_{i}} ∘ ⋯ ∘ f_{s_{1}}(K̂_{k}) > ϵ, diam f_{s_{i+1}} ∘ ⋯ ∘ f_{s_{1}}(K̂_{k}) ≤ ϵ.

We may assume i is the largest one satisfying the above. Then taking ϵ > 0 small enough, since the cardinality of generator system of G is finite we see that f_{s_{i}} ∘ ⋯ ∘ f_{s_{1}}(K̂_{k}) is simply connected. Set K̃_{k} = f_{s_{i}} ∘ ⋯ ∘ f_{s_{1}}(K̂_{k}) and g̃_{k} = f_{s_{l}} ∘ ⋯ ∘ f_{s_{i+1}}. Since R̂_{k}^{γ} − R̂_{k} does not contain any critical values of g_{k}|_{V_{k}}, the element K̃_{k}^{+} ∈ c(R̂_{k}^{γ}, g̃_{k}) containing K̃_{k} is also simply connected.

Since deg(g̃_{k}|K̃_{k}^{+}) ≤ 1 + N and diam K̃_{k} > ϵ, by Corollary 1.9 we see that there exists a positive real number r such that for each k,

B(z_{k}, r) ⊂ K̃_{k},

for some z_{k} ∈ \overline{ℂ}. We can assume that (z_{k}) converges to a point z ∈ \overline{ℂ} and (R̂_{k}) converges to a point y ∈ U. Then (g̃_{k}) is normal in B(z, r/2) and we can assume that (g̃_{k}) converges to y locally uniformly on B(z, r/2). But this contradicts to the assumption 1. Hence the lemma holds.

Now, let t = ♯T, N = ( max_{j=1,... ,m} deg(f_{j}))^{t} and ϵ < \frac{ϵ̃}{36N} . We can assume

that ϵ̃ is sufficiently small and diam U ≤ ϵ̃. Let n_{0} ∈ ℕ be an integer in Lemma 3.3 for these ϵ and N.

Lemma 3.4. Let G be an element of the form f = f_{w_{1}} ∘ ⋯ ∘ f_{w_{k}}. Let B be a simply connected subdomain of U, B′ ∈ c(B, f) an element such that △(B′, f) > N. Then there exists some ν ∈ {0, …, k − 1} such that if we set B_{ν} = f_{w_{k−ν}} ∘ ⋯ ∘ f_{w_{k}}(B′), then B_{ν} is simply connected, diam (B_{ν}) ≥ ϵ̃, and

deg(f_{w_{1}} ∘ ⋯ f_{w_{k−ν−1}}|B_{ν} : B_{ν} → B) ≤ N.

Proof. Suppose diam B_{ν} < ϵ̃ for each ν = 1, … k − 1, then B′ is simply connected(Note that we can assume ϵ̃ is sufficiently small) and deg(f|_{B′} : B′ → B) ≤ N. Hence △(B′, f) ≤ N and it is a contradiction. Hence there exists a ν ∈ {1, …, k − 1} such that

diam B_{ν} ≥ ϵ̃.

Take the maximal ν(1 ≤ ν ≤ k − 1) satisfying the above. Then B_{ν} is simply connected and

deg(f_{w_{1}} ∘ ⋯ ∘ f_{w_{k−ν−1}}|_{B_{ν}} : B_{ν} → B) ≤ N.

Now we will show the Theorem 3.1.

Proof. Take ϵ̃, ϵ and N as before. Take n_{0} in Lemma 3.3 for ϵ and N. Let k be the smallest integer such that there exists some admissable square Q = Q_{μ,n} at level n ≥ n_{0} with diam (K) > ϵ for some element K of c(Q, f_{w_{1}} ∘ ⋯ ∘ f_{w_{k}}) where (w_{1}, …, w_{k}) is some word of length k. We have k ≥ 1. Let Q̃ be the square attached to Q. By lemma 3.3, there exists an element S ∈ c(Q̃, f_{w_{1}} ∘ ⋯ ∘ f_{w_{k}}) such that △(S, f_{w_{1}} ∘ ⋯ ∘ f_{w_{k}}) > N. Take an integer ν with 1 ≤ ν < k in Lemma 3.4. Then we have

diam (f_{w_{k−ν}} ∘ ⋯ ∘ f_{w_{k}}(S)) > ϵ̃.

If we set S̃ = f_{w_{k−ν}} ∘ ⋯ ∘ f_{w_{k}}(S) then

deg(f_{w_{1}} ∘ ⋯ ∘ f_{w_{k−ν−1}}|S̃) ≤ N

and

S̃ ⊂ ⋃_{μ}(f_{w_{1}} ∘ ⋯ ∘ f_{w_{k−ν−1}})^{−1}(Q_{μ, n+1}).

By the minimality of k, we have that the diameter of each element of c(Q_{μ, n+1}, f_{w_{1}} ∘⋯∘f_{w_{k−ν−1}}) is less than ϵ. Since deg(f_{w_{1}} ∘⋯∘f_{w_{k−ν−1}}|S̃) ≤ N, we have that

ϵ̃ < diam S̃ ≤ 36Nϵ.

This contradicts to ϵ < \frac{ϵ̃}{36N} . Hence we have proved that for each admissable square Q_{μ,n} with n ≥ n_{0} and each g ∈ G, each element K ∈ c(Q_{μ,n}, g) satisfies that diam (K) < ϵ. Since ϵ is sufficiently small, K is simply connected. By Lemma 3.4, we have that

deg(f|_{K} : K → Q_{μ,n}) ≤ N + 1.

Hence z_{0} ∈ SH_{N+1}.

Now we will show the Corollary 3.2.

Proof. If we assume the conditions in the assumption of Corollary 3.2, then clearly the conditions in the assumption of Theorem 3.1 are satisfied. Hence we have z_{0} ∈ SH_{N}(G) for some N ∈ ℕ. Now take a small disc W around z_{0} contained in SH_{N}(G) ∩ U_{2}. If there exists a constant C > 0, a sequence (g_{n}) ⊂ G consisting of mutually distinct elements and a sequence (W_{n}) with W_{n} ∈ c(W, g_{n}) such that diam W_{n} > C for each n, then by Corollary 1.9, there exists a positive real number r such that for each n, we have B(z_{n}, r) ⊂ W_{n} for some z_{n} ∈ \overline{ℂ}. We can assume (z_{n}) converges to a point y ∈ \overline{ℂ}. Then (g_{n}) is normal in B(y, r/2). Since g_{n}(B(y, r/2)) ⊂ W ⊂ U_{1} for each large n ∈ ℕ, we get a contradiction. Hence the statement of the Corollary holds.

Theorem 3.5. Let G = ⟨f_{1}, f_{2}, … f_{n}⟩ be a finitely generated rational semigroup. Assume that there exists an element of G with the degree at least two, that each element of Aut \overline{ℂ} ∩ G(if this is not empty) is loxodromic and that F (G) ≠ ∅. Then G is semi-hyperbolic if and only if all of the following conditions are satisfied.

1. for each z ∈ J(G) there exists a neighborhood U of z in \overline{ℂ} such that for any sequence (g_{n}) ⊂ G, any domain V in \overline{ℂ} and any point ζ ∈ U, we have that the sequence (g_{n}) does NOT converge to ζ locally uniformly on V 2. for each j = 1, …, m each c ∈ C(f_{j}) ∩ J(G) satisfies

d(c, (G ∪ {id})(f_{j}(c))) > 0

Proof. First assume the conditions 1 and 2. Then by Theorem 3.1, we have that G is semi-hyperbolic.

Conversely, suppose G is semi-hyperbolic. By Lemma 1.13, the condition 2. holds. Now we will show the condition 1.holds. By Theorem 1.35, there exists an attractor K in F (G) for G. Let z_{0} be any point and U a neighborhood of z_{0} such that \overline{U} ∩ K = ∅. Suppose that there exists a sequence (g_{n}) ⊂ G, a domain V in \overline{ℂ} and a point ζ ∈ U such that g_{n} → ζ as n → ∞ locally uniformly on V. We will deduce a contradiction. We can assume that there exists a word w ∈ {1, …, m}^{ℕ} such that for each n,

g_{n} = α_{n} ∘ f_{w_{n}} ∘ ⋯ ∘ f_{w_{1}},

where α_{n} ∈ G is an element. Then from Theorem 1.35 and that \overline{U} ∩ K = ∅, we have that

f_{w_{n}} ∘ ⋯ ∘ f_{w_{1}}(V) ⊂ J(G),  (20)

for each n. Hence (f_{w_{n}} ∘ ⋯ ∘ f_{w_{1}})_{n} is normal in V. Now let us consider the rational skew product f̃ constructed by the generator system {f_{1}, …, f_{m}}. By the second statement of Theorem 2.13, we have that {w} × V ⊂ F̃(f̃). Hence there exists a positive integer n such that f_{w_{n}} ∘ ⋯ ∘ f_{w_{1}}(V) ⊂ F (G), if we take V sufficiently small. But this contradicts to (20). Hence we have shown that the condition 1. holds.

Theorem 3.6. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated sub-hyperbolic rational semigroup. Assume that there exists an element of G with the degree at least two, that each element of Aut \overline{ℂ} ∩ G(if this is not empty) is loxodromic and that there is no super attracting fixed point of any element of G in J(G). Then there exists a Riemannian metric ρ on a neighborhood V of J(G)\P (G) such that for each z_{0} ∈ J(G)\G^{−1}(P (G)∩J(G)), if there exists a word w = (w_{1}, w_{2}, …,) ∈ {1, …, m}^{ℕ} satisfying (f_{w_{n}} ⋯ f_{w_{1}})(z_{0}) ∈ J(G) for each n, then

∥(f_{w_{n}} ⋯ f_{w_{1}})′(z_{0})∥ → ∞, as n → ∞,

where ∥ · ∥ is the norm of the derivative measured from ρ on V to it.

Proof. By Theorem 1.36, there exists an attractor K in F (G) for G such that K^{i} ⊃ P (G) ∩ F (G). Let {V_{1}, …, V_{t}} be the set of all connected components of \overline{ℂ} \ K having non-empty intersection with J(G). We take the hyperbolic metric in V_{i} \ P (G) for each i = 1, …, t. We denote by ρ the Riemannian metric in V = ∪_{i=1}^{t}V_{i} \ P (G). First we will show the following.

• Claim 1. there exists a k ∈ ℕ such that for each n,

∥(f_{w_{n+k}} ⋯ f_{w_{n}})′(f_{w_{n}} ⋯ f_{w_{1}}(z_{0}))∥ > 1,

where ∥ · ∥ is the norm of the derivative measured from ρ to it. For each i = 1, …, t, let x_{i} be a point of V_{i} ∩ F (G). Since K is an attractor in F (G) for G, there exists a k ∈ ℕ such that for each (i_{1}, …, i_{k}) ∈ {1, …, m}^{k},

(f_{i_{k}} ⋯ f_{i_{1}})(x_{i}) ∈ K, for i = 1, …, t.  (21)

Let x be a point of J(G) ∩ V_{i} \ P (G). Suppose (f_{i_{k}} ⋯ f_{i_{1}})(x) ∈ V_{j} \ P (G) for some (i_{1}, …, i_{k}) ∈ {1, …, m}^{k} and j. Let U be the connected component of (f_{i_{k}} ⋯ f_{i_{1}})^{−1}(V_{j} \ P (G)) ∩ (V_{i} \ P (G)) containing x. Then

(f_{i_{k}} ⋯ f_{i_{1}}) : U → V_{j} \ P (G)

is a covering map. Hence we have

∥(f_{i_{k}} ⋯ f_{i_{1}})′(z)∥_{U, V_{j}\P (G)} = 1, for each z ∈ U,  (22)

where we denote by ∥ · ∥_{U, V_{j}\P (G)} the norm of the derivative measured from the hyperbolic metric on U to that on V_{j} \ P (G). On the other hand, by (21), U ≠ V_{i} \ P (G). Therefore the inclusion map i : U → V_{i} \ P (G) satisfies that

∥i′(z)∥_{U, V_{i}\P (G)} < 1, for each z ∈ U,  (23)

where we denote by ∥ · ∥_{U, V_{i}\P (G)} the norm of the derivative measured from the hyperbolic metric on U to that on V_{i} \ P (G). By (22) and (23), we get

∥(f_{i_{k}} ⋯ f_{i_{1}})′(z)∥_{V_{i}\P (G), V_{j}\P (G)} > 1, for each z ∈ U,  (24)

where we denote by ∥ · ∥_{V_{i}\P (G), V_{j}\P (G)} the norm of the derivative measured from the hyperbolic metric on V_{i} \ P (G) to that on V_{j} \ P (G). Hence the Claim 1. holds.

By Claim 1., we get that if the sequence (f_{w_{n}} ⋯ f_{w_{1}})(z_{0}))_{n=1}^{∞} does not accumulate to any point of P (G) ∩ J(G), then ∥(f_{w_{n}} ⋯ f_{w_{1}})′(z_{0})∥ → ∞ as n → ∞. Hence we can assume that the sequence accumulates to a point of P (G) ∩ J(G). We set

g_{n} = f_{w_{nk}} ⋯ f_{w_{1}}, for each n.

We will show the following.

• Claim 2. ∥(g_{n})′(z_{0})∥ → ∞ as n → ∞.

Since z_{0} ∈ J(G) \ G^{−1}(P (G) ∩ J(G)), by the same arguments as that in the proof of Theorem 1.34, we can show that there exists an ϵ_{1} > 0 and a sequence (n_{j}) of integers such that

g_{n_{j}}(z_{0}) ∈ J(G) \ B(P (G), ϵ_{1}), g_{n_{j}+1}(z_{0}) ∈ J(G) ∩ B(P (G), ϵ_{1}).

Suppose the case there exists a constant ϵ_{2} such that for each j,

d(g_{n_{j}+1}(z_{0}), P (G)) ≥ ϵ_{2}.

Then from Claim 1, there exists a constant c > 1 such that for each j,

∥(f_{w_{(n_{j} +1)k}} ⋯ f_{w_{n_{j} k+1}})′((f_{w_{n_{j} k}} ⋯ f_{w_{1}})(z_{0}))∥ > c.

Using the Claim 1 again, we can show that ∥(g_{n})′(z_{0})∥ → ∞ as n → ∞.

Next suppose the case there exists a subsequence (h_{l})_{l=1}^{∞} of (g_{n_{j}+1})_{j=1}^{∞} such that d(h_{l}(z_{0}), P (G)) → 0 as l → ∞. There exists a subsequence (β_{l})_{l=1}^{∞} of (g_{n_{j}})_{j=1}^{∞} such that for each l h_{l} = α_{l} ∘ β_{l} where α_{l} is an element of G. Then there exists a constant c_{1} ∈ ℕ such that for each l, wl_{S}(α_{l}) ≤ c_{1} where S = {f_{1}, …, f_{m}}. Hence there exists a sequence (x_{l}) such that d(x_{l}, β_{l}(z_{0})) → 0 as l → ∞ and α_{l}(x_{l}) ∈ P (G) for each l ∈ ℕ. We can assume that x_{l} ∈ B(β_{l}(z_{0}), ϵ_{1}) for each l ∈ ℕ. Let γ_{l} be the analytic inverse branch of β_{l} in B(β_{l}(z_{0}), ϵ_{1}) such that

γ_{l}(β_{l}(z_{0})) = z_{0}, for each l ∈ ℕ.

Since ∪_{l=1}^{∞}γ_{l}(B(β_{l}(z_{0}), ϵ_{1})) ⊂ \overline{ℂ} \ K and d(x_{l}, β_{l}(z_{0})) → 0, We get γ_{l}(x_{l}) → z_{0} as l → ∞. Hence we have

d(z_{0}, h_{l}^{−1}(P (G))) → 0, as l → ∞.  (25)

There exists an i such that z_{0} ∈ V_{i} \ P (G). For each l let V_{j_{l}} be the element of {V_{1}, …, V_{t}} such that h_{l}(z_{0}) ∈ V_{j_{l}} \ P (G). Let W_{l} be the connected component of h_{l}^{−1}(V_{j_{l}} \ P (G)) ∩ V_{i} \ P (G) containing z_{0}. Then h_{l} : W_{l} → V_{j_{l}} is a covering map. Hence we have

∥(h_{l})′(z)∥_{W_{l}, V_{j_{l}}\P (G)} = 1, for z ∈ W_{l},

where ∥ · ∥_{W_{l}, V_{j_{l}}\P (G)} is the norm of the derivative measured from the hyperbolic metric on W_{l} to that on V_{j_{l}}. By Theorem 2.25 in [M], (25) implies that

∥(i_{l})′(z)∥_{W_{l}, V_{i}\P (G)} → 0 as l → ∞,

where we denote by i_{l} the inclusion map from W_{l} into V_{i} \ P (G) for each l ∈ ℕ. It follows that

∥h′_{l}(z)∥_{V_{i}\P (G), V_{j_{l}}\P (G)} → ∞ as l → ∞,  (26)

where ∥ · ∥_{V_{i}\P (G), V_{j_{l}}\P (G)} is the norm of the derivative measured from the hyperbolic metric on V_{i} \ P (G) to that on V_{j_{l}} \ P (G). By (26) and Claim 1, we get ∥(g_{n})′(z_{0})∥ → ∞ as n → ∞. Hence the Claim 2 holds.

In the same way we can show that for each i = 1, …, k − 1,

∥(f_{w_{nk+i}} ⋯ f_{w_{1}})(z_{0})∥ → ∞ as n → ∞.

We have thus proved the Theorem.

Theorem 3.7. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated sub-hyperbolic rational semigroup. Assume that there exists an element of G with the degree at least two, that each element of Aut \overline{ℂ} ∩ G(if this is not empty) is loxodromic and that there is no super attracting fixed point of any element of G in J(G). Then G is semi-hyperbolic.

Proof. We will appeal to Theorem 3.5. Since there is no super attracting fixed point of any element of G in J(G), the condition 2. in Theorem 3.5 is satisfied. By Theorem 1.36, there exists an attractor K in F (G) for G. Let z_{0} be any point and U a neighborhood of z_{0} such that \overline{U} ∩ K = ∅. Suppose that there exists a sequence (g_{n}) ⊂ G, a domain V in \overline{ℂ} and a point ζ ∈ U such that g_{n} → ζ as n → ∞ locally uniformly on V. We will deduce a contradiction. We can assume that there exists a word w ∈ {1, …, m}^{ℕ} such that for each n,

g_{n} = α_{n}f_{w_{n}} ∘ ⋯ ∘ f_{w_{1}},

where α_{n} ∈ G is an element. Then from Theorem 1.35 and that \overline{U} ∩ K = ∅, we have that

f_{w_{n}} ∘ ⋯ ∘ f_{w_{1}}(V) ⊂ J(G),  (27)

for each n. Hence (f_{w_{n}} ∘ ⋯ ∘ f_{w_{1}})_{n} is normal in V. Let z_{1} ∈ V ∩ G^{−1}(P (G) ∩ J(G)) be a point. By the backward self-similarity of J(G) and Lemma 1.33, there exists a sequence (n_{j}) of positive integers and a neighborhood W of P (G) ∩ J(G) in \overline{ℂ} such that for each j,

f_{w_{n_{j}}} ∘ ⋯ ∘ f_{w_{1}}(z_{1}) ∈ \overline{ℂ} \ W.

By Theorem 3.6, we have that

∥(f_{w_{n_{j}}} ∘ ⋯ ∘ f_{w_{1}})′(z_{1})∥ → ∞, as j → ∞,  (28)

where ∥ · ∥ denotes the norm of the derivative with respect to the spherical metric. Since (f_{w_{n}} ∘⋯∘f_{w_{1}})_{n} is normal in V, this is a contradiction. Hence the condition 1 in Theorem 3.5 is satisfied. By Theorem 3.5, we get that G is semi-hyperbolic.

4 Open Set Condition and Area 0

Definition 4.1. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated rational semigroup. We say that G satisfies the open set condition with respect to the generators f_{1}, f_{2}, …, f_{m} if there exists an open set O such that for each j = 1, …, m, f_{j}^{−1}(O) ⊂ O and {f_{j}^{−1}(O)}_{j=1,... ,m} are mutually disjoint.

Definition 4.2. Let G be a rational semigroup and S = {f_{λ} | λ ∈ Λ} a generator system of G. For each g ∈ G, We set

wl_{S}(g) = min{n ∈ ℕ | g = f_{λ_{1}} ⋯ f_{λ_{n}}}.

We call wl_{S}(g) the word length of g with respect to S.

Proposition 4.3. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated rational semigroup. Assume that G satisfies the open set condition with respect to the generators f_{1}, f_{2}, …, f_{m} and O \ J(G) ≠ ∅ where O is an open set in the definition of the open set condition. Then J(G)^{i} = ∅ where we denote by J(G)^{i} the interior of J(G).

Proof. Let S = {f_{1}, …, f_{m}}. Assume that J(G)^{i} ≠ ∅.

Then we claim that for each element g ∈ G and each point x ∈ J(G)^{i},

g(x) ∈ \overline{ℂ} \ (O \ J(G)).

Suppose that there exists a point y ∈ J(G)^{i} and an element g_{1} ∈ G such that g_{1}(y) ∈ O \ J(G). Since J(G) = ∪_{i=1}^{n}f_{i}^{−1}(J(G)), there exists an element h ∈ G with wl_{S}(h) = wl_{S}(g_{1}) such that h(y) ∈ J(G). Since f_{j}^{−1}(O) ⊂ O for each j = 1, …, m, we have J(G) ⊂ \overline{O} and J(G)^{i} ⊂ O. Hence g_{1}^{−1}(O) ∩ h^{−1}(O) ≠ ∅. But g_{1} ≠ h and that is a contradiction because of the open set condition. Therefore the above claim holds.

Now the claim implies that G is normal in J(G)^{i} but this is a contradiction and so we have J(G)^{i} = ∅.

Theorem 4.4. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated rational semigroup which is semi-hyperbolic, contains an element with the degree at least two and satisfies the open set condition with respect to the generators f_{1}, f_{2}, …, f_{m}. Let O be an open set in Definition 4.1. Assume that ♯(∂O ∩ J(G)) < ∞. Then the 2-dimensional Lebesgue measure of J(G) is equal to 0.

Proof. We will show the statement using the method of Theorem 1.3 in [Y]. We fix a gemerator system S = {f_{1}, …, f_{m}}. By the assumption of our Theorem, we have each element of Aut \overline{ℂ} ∩ G(if this is not empty) is loxodromic. By Theorem 1.35, A(G) is an attractor in F (G) for G. We can assume ∞ ∈ A(G). Suppose that the 2-dimensional Lebesgue measure of J(G) is positive.

Since ♯(∂O ∩ J(G)) < ∞, G^{−1}(G(∂O ∩ J(G))) is a countable set. Hence there exists a Lebesgue density point x of J(G) such that x ∈ J(G) \ (G^{−1}(G(∂O ∩ J(G))). Since we have J(G) = ∪_{j=1}^{m}f_{j}^{−1}(J(G)), there exists a word w = (w_{1}, w_{2}, …) ∈ {1, …, m}^{ℕ} such that for each positive integer u, f_{w_{u}} ⋯ f_{w_{1}}(x) ∈ J(G).

We will show that the sequence (f_{w_{u}} ⋯ f_{w_{1}}(x))_{u} has an accumulation point in J(G)\∂O. Assume that is false. For each large u, let ζ_{u} be the closest point to f_{w_{u}} ⋯ f_{w_{1}}(x) in ∂O∩J(G). Since there exists no super attracting fixed point of any point of any element of G in J(G), there exists a positive integer s such that for each integer t with t ≥ s, (f_{w_{t}} ⋯ f_{w_{s}})′(ζ_{s−1}) ≠ 0. Since G is semi-hyperbolic, we have that for each x ∈ ∂O ∩ J(G), if there exists an element g ∈ G such that g(x) = x, then x is a repelling fixed point of g. Applying Lemma 1.33, we get a contradiction. Hence the sequence (f_{w_{u}} ⋯ f_{w_{1}}(x))_{u} has an accumulation point in J(G) \ ∂O.

By the argument above, we have that there exists an ϵ > 0 and a sequence (g_{n}) of elements of G such that for each n, g_{n+1} = h_{n}g_{n} for some h_{n} ∈ G and g_{n}(x) ∈ J(G) \ D(∂O, ϵ). Let δ be a small number so that δ < ϵ and for each g ∈ G and each x ∈ J(G),

deg(g : U → D(x, δ)) ≤ N

for each U ∈ c(D(x, δ), g), where N is a positive integer independent of x, g and U. By Lemma 1.12, we can assume that for each g ∈ G and each x ∈ J(G), if V is a simply connected open neighborhood of x contained in D(x, δ), then each element of c(D(x, δ), g) is simply connected.

For each n, we set x_{n} = g_{n}(x). Let U_{n} be the conncted component of g^{−1}(D(x_{n}, \frac{1}{2}δ)) containing x. Now we will claim that

lim_{n→∞} \frac{m_{2}(U_{n} ∩ J(G))}{m_{2}(U_{n})} = 1,  (29)

where we denote by m_{2} the 2-dimensional Lebesgue measure. By Corollary 1.9, Proposition 4.3 and Corollary 2.15, there exist a constant K > 0, two sequences (r_{n}) and (R_{n}) such that \frac{1}{K} ≤ \frac{r_{n}}{R_{n}} < 1, R_{n} → 0 and

D(x, r_{n}) ⊂ U_{n} ⊂ D(x, R_{n}).

Since x is a Lebesgue density point of J(G), the claim holds. Now we get

lim_{n→∞} \frac{m_{2}(U_{n} ∩ F (G))}{m_{2}(U_{n})} = 0.  (30)

For each n, Let ϕ_{n} : D(0, 1) → D_{g_{n}}(x_{n}, δ) be the Riemann map such that ϕ_{n}(0) = x, where D_{g_{n}}(x_{n}, δ) is the element of c(D(x_{n}, δ), g_{n}) containing U_{n}. By (30) and the Koebe distortion theorem, we get

lim_{n→∞} \frac{m_{2}(ϕ_{n}^{−1}(U_{n} ∩ F (G)))}{m_{2}(ϕ_{n}^{−1}(U_{n}))} = 0.  (31)

By Corollary 1.8, there exists a constant 0 < c_{1} < 1 such that for each n, the Euclidian diameter of ϕ_{n}^{−1}(U_{n}) is less than c_{1}. Since we can assume that D_{g_{n}}(x_{n}, δ) ⊂ ℂ for each n and uniformly bounded in ℂ, by Cauchy’s formula, we get that there exists a constant c_{2} such that

|(g_{n}ϕ_{n})′(z)| ≤ c_{2} on ϕ_{n}^{−1}(U_{n}), n = 1, 2, …  (32)

Now we will show

D(x_{n}, \frac{1}{2}δ) ∩ F (G) = g_{n}(U_{n} ∩ F (G)), for each n.  (33)

It is easy to see that D(x_{n}, \frac{1}{2}δ)∩F (G) ⊃ g_{n}(U_{n}∩F (G)). Now let z be a point of D(x_{n}, \frac{1}{2}δ) ∩ F (G) and assume that there exists a point w ∈ U_{n} ∩ J(G) such that g_{n}(w) = z. Since J(G) = ∪_{j=1}^{m}f_{j}^{−1}(J(G)) and g_{n}(w) ∈ F (G), there exists an element g ∈ G with wl_{S}(g) = wl_{S}(g_{n}) such that g(w) ∈ J(G) ⊂ \overline{O}. Hence we have g ≠ g_{n} and g^{−1}(O) ∩ g_{n}^{−1}(O) ≠ ∅. But this contradicts to the open set condition. Therefore (33) holds.

By (31) , (32) and (33), we have

\frac{m_{2}(D(x_{n}, \frac{1}{2}δ) ∩ F (G))}{m_{2}(D(x_{n}, \frac{1}{2}δ))} = \frac{m_{2}((g_{n} ∘ ϕ_{n})(ϕ_{n}^{−1}(U_{n} ∩ F (G)))}{m_{2}(D(x_{n}, \frac{1}{2}δ))}

≤ \frac{∫_{ϕ_{n}^{−1}(U_{n}∩F (G))} |(g_{n} ∘ ϕ_{n})′(z)|^{2}dm_{2}(z)}{m_{2}(ϕ_{n}^{−1}(U_{n}))} \frac{m_{2}(ϕ_{n}^{−1}(U_{n}))}{m_{2}(D(x_{n}, \frac{1}{2}))}
→ 0,

as n → ∞. Hence we have

lim_{n→∞} \frac{m_{2}(D(x_{n}, \frac{1}{2}δ) ∩ J(G))}{m_{2}(D(x_{n}, \frac{1}{2}δ))} = 1.

We can assume that there exists a point x_{∞} ∈ J(G) such that x_{n} → x_{∞}. Then

\frac{m_{2}(D(x_{∞}, \frac{1}{2}δ) ∩ J(G))}{m_{2}(D(x_{∞}, \frac{1}{2}δ))} = 1.

This implies that D(x_{∞}, \frac{1}{2}δ) ⊂ J(G) but this is a contradiction because we have J(G)^{i} = ∅ by Proposition 4.3.

Corollary 4.5. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated rational semigroup which is sub-hyperbolic, contains an element with the degree at least two and satisfies the open set condition with respect to the generators f_{1}, f_{2}, …, f_{m}. Let O be an open set in Definition 4.1. Assume that ♯(∂O ∩ J(G)) < ∞. Then the 2-dimensional Lebesgue measure of J(G) is equal to 0.

Proof. By Proposition 4.3, J(G)^{i} = ∅. Since G is finitely generated, by [HM2], there is no super attracting fixed point of any element of G in ∂J(G) = J(G). Therefore by Theorem 3.7, we have that G is semi-hyperbolic. By Theorem 4.4, the statement holds.

5 δ-subconformal measure and Hausdorff dimension of Julia sets

Definition 5.1. 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 if for each g ∈ G and for each Borel measurable set A

μ(g(A)) ≤ ∫_{A} ∥g′(z)∥^{δ}dμ,

where we denote by ∥ · ∥ the norm of the derivative with respect to the spherical metric. 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 not s such that S(s, x) < ∞, then we set S(x) = ∞.Also we set

s_{0}(G) = inf{S(x)}, s(G) = inf{δ | ∃μ : δ-subconformal measure}

It is not difficult for us to prove the next result using the same method as that in [Sul].

Theorem 5.2 ([S2]). Let G be a rational semigroup which has at most countably many elements. If there exists a point x ∈ \overline{ℂ} such that S(x) < ∞ then there is a S(x)-subconformal measure. In particular, we have s(G) ≤ s_{0}(G).

Proposition 5.3 ([S2]). Let G be a rational semigroup and τ a δ-subconformal measure for G where δ is a real number. Assume that ♯J(G) ≥ 3 and for each x ∈ E(G) there exists an element g ∈ G such that g(x) = x and |g′(x)| < 1. Then the support of τ contains J(G).

Proposition 5.4. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated rational semigroup. Assume that G satisfies the open set condition with respect to the generators f_{1}, f_{2}, …, f_{m} and O \ J(G) ≠ ∅ where O is an open set in the definition of the open set condition. If there exists an attractor in F (G) for G, then

s_{0}(G) ≤ 2.

Proof. We can assume m ≥ 2. Let K be an attractor in F (G) for G. There exists a simply connected domain U in (O ∩ F (G)) \ (K ∪ P (G)) such that g(U) ∩ U = ∅ for each g ∈ G. By the open set condition, it is easy to see that if g ≠ h, then g^{−1}(U) ∩ h^{−1}(U) = ∅. Hence we have

∑_{S} ∫_{U} ∥S′(z)∥^{2}dm_{2}(z) < ∞,

where S is taken over all holomorphic inverse branches of all elements of G defined on U, ∥ · ∥ denotes the norm of the derivative with respect to the spherical metric and m_{2} is the 2-dimensional Lebesgue measure on \overline{ℂ}. It follows that for almost every where x ∈ U, S(2, x) < ∞.

Lemma 5.5. Let G be a rational semigroup. Assume that ∞ ∈ F (G), ♯J(G) ≥ 3 and for each x ∈ E(G) there exists an element g ∈ G such that g(x) = x and |g′(x)| < 1. We also assume that there exist a countable set E in \overline{ℂ}, positive numbers a_{1} and a_{2} and a constant c with 0 < c < 1 such that for each x ∈ J(G) \ E, there exist two sequences (r_{n}) and (R_{n}) of positive real numbers and a sequence (g_{n}) of 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}).

Let δ be a real number with δ ≥ s(G) and μ a δ- subconformal measure. Then δ-Hausdorff measure on J(G) is absolutely continuous with respect to μ such that the Radon-Nikodim derivative is bounded from above. In particular, we have

dim_{H}(J(G)) ≤ s(G).

Proof. By Proposition 5.3, the support of μ contains J(G). Hence there exists a constant c_{1} > 0 such that for each x ∈ J(G), μ(D(x, a_{2})) > c_{1}.

Fix any x ∈ J(G) \ E. For each n we set R̃_{n}(z) = R_{n}z + x. By the condition 1 and 2, the family {g_{n} ∘ R̃_{n}} is normal in D(0, 1). By Marty’s theorem, there exists a constant c_{2} such that for each n and each w ∈ D(0, c),

∥(g_{n} ∘ R̃_{n})′(w)∥ ≤ c_{2}.

Note that we can take the constant c_{2} independent of x ∈ J(G) \ E. Hence we have for each n,

c_{1} ≤ μ(D(g_{n}(x), a_{2}))
≤ μ(g_{n}(D(x, r_{n})))
≤ ∫_{D(x,r_{n})} ∥g′_{n}(z)∥^{δ}dμ(z)
= ∫_{D(x,r_{n})} ∥(g_{n} ∘ R̃_{n} ∘ R̃_{n}^{−1})′(z)∥^{δ}dμ(z)
≤ c_{3} \frac{1}{R_{n}^{δ}} μ(D(x, r_{n}))
≤ c_{3} \frac{1}{r_{n}^{δ}} μ(D(x, r_{n})),

where c_{3} is a constant not depending on n and x ∈ J(G) \ E. Therefore we get that there exists a constant c_{4} not depending on n and x ∈ J(G) \ E such that

\frac{μ(D(x, r_{n}))}{r_{n}^{δ}} ≥ c_{4}.  (34)

Now we can show the statement of our lemma in the same way as the proof of Theorem 14 in [DU]. We will follow it. Let A be any Borel set in J(G). We set A_{1} = A \ E. We denote by H_{δ} the δ-Hausdorff measure. Since E is a countable set, we have H_{δ}(A) = H_{δ}(A_{1}). Fix γ, ϵ. For every x ∈ A_{1}, denote by {r_{n}(x)}_{j=1}^{∞} the sequence constructed in the above paragraph. Since μ is regular, for every x ∈ A_{1} there exists a radius r(x) being of the form r_{n}(x) such that

μ( ⋃_{x∈A_{1}} D(x, r(x)) \ A_{1}) < ϵ.

By the Besicovič theorem we can choose a countable subcover {D(x_{i}, r_{x_{i}})}_{i=1}^{∞} from the cover {D(x, r(x)}_{x∈A_{1}} of A_{1}, of multiplicity bounded by some constant C ≥ 1, independent of the cover. By (34), we obtain

∑_{i=1}^{∞} r(x_{i})^{δ} ≤ c_{4}^{−1} ∑_{i=1}^{∞} μ(D(x_{i}, r(x_{i})))

≤ c_{4}^{−1}Cμ(⋃_{i=1}^{∞} D(x_{i}, r(x_{i})))
≤ c_{4}^{−1}C(ϵ + μ(A_{1})).

Letting ϵ → 0 and then γ → 0 we get

H_{δ}(A) = H_{δ}(A_{1}) ≤ c_{4}^{−1}Cμ(A_{1}) ≤ c_{4}^{−1}Cμ(A).

Theorem 5.6. Let G be a rational semigroup generated by a generator system \{f_{λ}\}_{λ∈Λ} such that ∪_{λ∈Λ}\{f_{λ}\} is a compact subset of End(\overline{ℂ}). Let f̃ be a rational skew product constructed by the generator system. Assume f̃ is semi-hyperbolic along fibres and satisfies the condition C1 with a family of discs {D_{x}}_{x∈X} such that D_{x} = D, ∀x ∈ X with some D. Then we have

dim_{H}(J(G)) ≤ s(G).

Proof. We can assume ∞ ∈ F (G). Let x be any point of J(G). Since we have J(G) = ∪_{λΛ}f_{λ}^{−1}(J(G)), for each n ∈ ℕ there exists an element g_{n} ∈ G which is a product of n generators such that g_{n}(x) ∈ J(G). Let δ be a small positive number. For each n, we denote by D_{g_{n}}(g_{n}(x), δ) the element of c(D(g_{n}(x), δ), δ) containing x. By Theorem 2.13, if we take a δ smaller, then

diam (D_{g_{n}}(g_{n}(x), δ)) → 0, as n → ∞.  (35)

By Lemma 2.14, we can assume that D_{g_{n}}(g_{n}(x), δ) is simply connected for each n. Let ϕ_{n} : D(0, 1) → D_{g_{n}}(g_{n}(x), δ) be the Riemann map such that ϕ_{n}(0) = x. By the Koebe distortion theorem, we have that for each n,

D_{g_{n}}(g_{n}(x), δ) ⊃ D(x, \frac{1}{4}|ϕ′_{n}(0)|).

Since G is semi-hyperbolic, we can assume that D(J(G), δ) ⊂ SH_{N}(G) where N is a positive integer. By Corollary 1.10, we get

sup_{n∈ℕ}{diam (ϕ_{n}^{−1}(D_{g_{n}}(g_{n}(x), ϵδ)))} → 0, as ϵ → 0.

Therefore by the Koebe distortion theorem, there exists an ϵ such that

D_{g_{n}}(g_{n}(x), ϵδ) = ϕ_{n}(ϕ_{n}^{−1}(D_{g_{n}}(g_{n}(x), ϵδ)))
⊂ D(x, \frac{1}{8}|ϕ′_{n}(0)|), for each n.

By (35), we have |ϕ′_{n}(0)| → 0 as n → ∞. Applying Lemma 5.5, we get

dim_{H}(J(G)) ≤ s(G).

Theorem 5.7. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated rational semigroup which is semi-hyperbolic. Assume that G contains an element with the degree at least two, each element of Aut \overline{ℂ} ∩ G(if this is not empty) is loxodromic and F (G) ≠ ∅. Then we have

dim_{H}(J(G)) ≤ s(G) ≤ s_{0}(G).

Proof. By Theorem 5.6 and Theorem 5.2.

Remark 6. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated hyperbolic rational semigroup which satisfies the strong open set condition (i.e. G satisfies the open set condition with an open set O satisfying O ⊃ J(G).). We assume that when n = 1 the degree of f_{1} is at least two. By the results in [S4](Theorem 3.2 and the proof, Theorem 3.4 and Corollary 3.5), we have

0 < dim_{H} J(G) = s(G) = s_{0}(G) < 2.

Example 5.8. Let G = ⟨f_{1}, f_{2}⟩ where f_{1}(z) = z^{2} + 2, f_{2}(z) = z^{2} − 2. Since P (G) ∩ J(G) = {2, −2} and P (G) ∩ F (G) is compact, we have G is sub-hyperbolic. By Theorem 3.7, G is also semi-hyperbolic. Since f_{j}^{−1}(D(0, 2)) ⊂ D(0, 2) for j = 1, 2 andf_{1}^{−1}(D(0, 2)) ∩ f_{2}^{−1}(D(0, 2)) = ∅, G satisfies the open set condition. Also J(G) is included in B = ∪_{j=1}^{2}f_{j}^{−1}(\overline{D(0, 2)}). Since B ∩ ∂D(0, 2) = {2, −2, 2i, −2i}, we get ♯(J(G) ∩ ∂D(0, 2)) < ∞. By Corollary 4.4, we have m_{2}(J(G)) = 0, where we denote by m_{2} the 2dimensional Lebesgue measure. By Theorem 5.7 and Proposition 5.4, we have also

dim_{H}(J(G)) ≤ s(G) ≤ s_{0}(G) ≤ 2.

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