Dynamics of Sub-hyperbolic and Semi-hyperbolic Rational Semigroups ^{∗}

Hiroki Sumi ^{†}

Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 6068501, Japan e-mail; sumi@math.h.kyoto-u.ac.jp

May 18, 1998

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 functions. 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.

∗1991 Mathematical Subject Classification: 30D05, 58F23

†Partially supported by JSPS Research Fellowships for Young Scientists.

J(G) is backward invariant under G but not forward invariant in general. If G = ⟨f_{1}, f_{2}, …, f_{n}⟩ is a finitely generated rational semigroup , then J(G) has the backward self-similarity. That is, we have J(G) = ∪_{i=1}^{n}f_{i}^{−1}(J(G)). 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 detail about these properties, see [ZR], [GR], [HM1], [HM2], [S1] and [S2]. In this paper we use the notations in [HM1] and [S3].

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. By using these theorems, we can show the continuity of the Julia set with respect to the perturbation of the generators. By the existence of an attractor, we can also show the contracting property with respect to the backward dynamics(section2). Using that property, we will show that if a finitely generated rational semigroup G is semi-hyperbolic and satisfies the open set condition with the open set O such that ♯(∂O ∩ J(G)) < ∞ , then 2-dimensional Lebesgue measure of the Julia set is equal to 0(section2).

In the section 3, we will consider constructing δ-subconformal measures. If a rational semigroup has at most countably many elements, then we can construct δ-subconformal measures. We will see that if G is a finitely generated sub-hyperbolic rational semigroup and has no superattracting fixed point of any element of it in the Julia set, or if G is a finitely generated semi-hyperbolic rational semigroup and the interior of the Julia set is empty, then the Hausdorff dimension of the Julia set is less than the exponent δ. To show those results, the contracting property of backward dynamics will be used.

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

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({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 sub-hyperbolic and for each g ∈ G, there is no super attracting fixed point of g in J(G), 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 no indifferent cycles, Siegel disks and Hermann rings.

Lemma 1.6 ([CJY], [Y]). 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 1.7 ([Y]). 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 1.8 ([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.

Lemma 1.9. 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 U′ \ \overline{V′} 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 U′_{i} \ \overline{U′_{i+1}} ≥ \frac{1}{2πN} log \frac{r_{i+1}}{r_{i}} , for i = 0, …, n − 1.

It follows that

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

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

Lemma 1.10. 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 1.9, 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 1.11. 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(g(c))}.

Proof. Assume that there exists a critical point c of an element g ∈ G such that c ∈ SH_{N}(G) ∩ \overline{G(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 1.12. 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 1.11, we can show the statement immediately.

Definition 1.13. 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 1.14. 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 exist 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 1.15. 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 1.16. 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 exist 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 1.9, 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 1.17. Let G be a polynomial semigroup. Assume that there exist 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 1.16, 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 1.16, y ∈ UH(G).

Now we consider the case ♯(UH(G)) = 1 or 2. Then ∞ ∈ UH(G). If ∞ ∈ F (G), then since G(∞) = {∞}, from Lemma 1.16 the condition 3. holds. Now suppose ∞ ∈ J(G). There exists an element g ∈ G with the degree at least two. From Corollary 1.12, 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 exist 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 1.16. 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 1.18. 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 exist 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 [S3], we can show that there exist 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 1.14.

Theorem 1.19. 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 exist 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 1.17, we can show the statement in the same way as the proof of Theorem 1.18.

By Lemma 1.9 and using the method of the proof in Lemma 1.16, we can show the next lemma immediately.

Lemma 1.20. 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 1.21. Let G be a rational semigroup. Assume that there exist 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 1.17.

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 1.20, 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 1.20, 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 1.21, we can show the next result in the same way as the proof of Theorem 1.18.

Theorem 1.22. 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 exist 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 1.20, we can show the next result immediately.

Theorem 1.23. 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 1.21.

Theorem 1.24. 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 1.25. 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 1.26. 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 1.27. 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 1.26.

Lemma 1.28. 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 Möbius transformation in G 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 exist 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 exist 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 1.29. 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 1.30. 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 nonrepelling 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 1.31. 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 most close 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 exist 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 1.30, 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 1.30 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 1.32. 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 Möbius transformation in G 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.

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 1.26, 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 exist 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 exist 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 1.28, 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 1.22, we can show that (β_{n}(z)) accumulates only in ∂J(G) ∩ UH(G). For each large n, let ζ_{n} be the most close 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 1.31, 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 1.28, 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.

Finally 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.

By Theorem 1.32 and Lemma 1.27, we get the next theorem.

Theorem 1.33. 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 Möbius transformation in G 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.

Theorem 1.34. 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 Möbius transformation in G 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.

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 1.32, the statement of our theorem holds.

Proposition 1.35. 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 1.36. 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 1.32 and Theorem 2.3.4 in [S3], we get the following result.

Corollary 1.37. 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 1.32. Then the map

a ↦ J(G_{a})

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

Corollary 1.38. 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 Möbius transformation in G_{b} 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 Open Set Condition and Area 0

Definition 2.1. 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 2.2. 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 2.3. 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 exist 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} = ∅.

Lemma 2.4. Let V and W be simply connected domains in \overline{C}. Suppose that \overline{W} ⊂ V and mod (V \ \overline{W}) > 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 2.5. 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 Möbius transformation in G is loxodromic. Also assume that J(G)^{i} = ∅. Then there exist a δ > 0, a constant L with L > 0 and a constant λ with 0 < λ < 1 such that

sup\{diam_{S}U | U ∈ c(B(x, \frac{1}{2}δ), f_{i_{n}} ⋯ f_{i_{1}}), x ∈ J(G), (i_{1}, …, i_{n}) ∈ \{1, …, m\}^{n}\}

≤ Lλ^{n}, for each n.

Proof. We will show the statement in the same way as the proof of Proposition 3.3 in [Y] or Theorem 2.1 in [CJY]. In the proof of them, it was used that for each rational map f and each open set V with V ∩J(⟨f⟩) ≠ ∅, there exists a positive integer n such that f^{n}(V) ⊃ J(⟨f⟩). But in the case of semigroups, it is not true in general that for each word w = (w_{1}, w_{2}, …) ∈ {1, …, m} and for each open set V with V ∩ J(G) ≠ ∅, there exists an n such that (f_{w_{n}} ⋯ f_{w_{1}})(V) ⊃ J(G). So here, we will use the fact that there exists an attractor in F (G) for G(Theorem 1.33). We can assume that ∞ ∈ F (G). We have only to show the statement of our Lemma with respect to the Euclidian diameter. Let δ > 0 be a small number so that for each g ∈ G and x ∈ J(G),

deg(g : U → D(x, δ)) ≤ N.

for each U ∈ c(D(x, δ), g). By Theorem 1.33, there exists a ball B included in F (G) such that \overline{G(B)} ⊂ F (G). Hence by Lemma 1.10, we can assume that for each g ∈ G and 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. First we claim that

sup\{diam U | U ∈ c(D(x, \frac{1}{2}δ), f_{i_{n}} ⋯ f_{i_{1}}), x ∈ J(G), (i_{1}, …, i_{n}) ∈ \{1, …, m\}^{n}\}

→ 0, as n → ∞. Suppose that is false. Then there exist a constant C with C > 0, a sequence (y_{k}) of points in J(G), a sequence (g_{k}) of elements of G and a sequence (U_{k}) with U_{k} ∈ c(D(y_{k}, \frac{1}{2}δ), g_{k}) such that g_{k} is a product of n_{k} generators, n_{k} → ∞, and

diam U_{k} ≥ C, for each k.

We can assume that (y_{k}) converges to a point y_{0} in J(G) and that there exists a sequence (x_{k})_{k} with x_{k} ∈ g_{k}^{−1}(y_{k}) ∩ U_{k} converging to a point z_{0} in J(G). By Corollary 1.8, there exists a number r_{0} > 0 such that for each large k,

U_{k} ⊃ D(z_{0}, r_{0}).

Hence

g_{n_{k}}(D(z_{0}, r_{0})) ⊂ D(y_{k}, \frac{1}{2}δ) ⊂ D(y_{0}, δ) ⊂ \overline{C} \ UH(G),  (4)

for each large k. Since J(G)^{i} = ∅, we have D(z_{0}, r_{0}) ∩ F (G) ≠ ∅. By Theorem 1.33, we get a contradiction. Hence the above claim holds.

By the above claim, there exists a positive integer n_{0} such that for each n with n ≥ n_{0}, for each element g ∈ G which is a product of n generators and for each point y ∈ J(G),

diam (U) ≤ \frac{1}{4}δ,  (5)

for each U ∈ c(D(y, \frac{1}{2}δ), g). Fix any positive integer k. Let w = (w_{1}, w_{2}, …) ∈ {1, …, m}^{N} be any word. Let (x_{n}) be a sequence such that f_{w_{n}}(x_{n}) = x_{n−1} for each n. For each j = 0, …, k, let W_{j} be the element of

c(D(x_{(k−j)n_{0}}, \frac{1}{2}δ), f_{w_{(k−j)n_{0}+1}} ⋯ f_{w_{kn_{0}}})

containing x_{kn_{0}}. By (5), we have

W_{0} ⊃ ⋯ ⊃ W_{k}.

Since \overline{D(J(G), δ)} ⊂ \overline{C} \ UH(G), there exists a positive integer N such that \overline{D(J(G), δ)} ⊂ SH_{n}(G). Then for each j = 1, …, k,

f_{w_{(k−j)n_{0}+1}} ⋯ f_{w_{kn_{0}}} : W_{j} → D(x_{(k−j)n_{0}}, \frac{1}{2}δ)

is a proper holomorphic map with the degree at most N. Since

f_{w_{(k−j)n_{0}+1}} ⋯ f_{w_{kn_{0}}}(W_{j+1})

is a connected component of (f_{w_{(k−j−1)n_{0}+1}} ⋯ f_{w_{kn_{0}}})^{−1}(D(x_{(k−j−1)n_{0}}, \frac{1}{2}δ)), which is included in D(x_{(k−j)n_{0}}, \frac{1}{4}δ) by (5), we have that for each j = 0, …, k − 1,

mod (W_{j} \ \overline{W_{j+1}}) ≥ \frac{1}{2πN} log 2.

By Lemma 2.4, there exists a λ with 0 < λ < 1 depending only on N such that

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

Hence we get that diam W_{k} ≤ λ^{k}diam D(x_{0}, \frac{1}{2}δ). Therefore the statement of our lemma holds.

Theorem 2.6. 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 2.1. 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 Möbius transformation in G is loxodromic. By Theorem 1.33, 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 most close 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 1.31, 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 exist 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 1.10, we can assume that for each g ∈ G and each x ∈ J(G), if V is a simply connected 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,  (6)

where we denote by m_{2} the 2-dimensional Lebesgue measure. By Corollary 1.8, Proposition 2.3 and Lemma 2.5, 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.  (7)

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 (7) 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.  (8)

By Corollary 1.7, 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 Caushy’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, …  (9)

Now we will show

D(x_{n}, \frac{1}{2}δ) ∩ F (G) = g_{n}(U_{n} ∩ F (G)), for each n.  (10)

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 (10) holds.

By (8) , (9) and (10), 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 2.3.

Corollary 2.7. 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 2.1. Assume that ♯(∂O ∩ J(G)) < ∞. Then the 2-dimensional Lebesgue measure of J(G) is equal to 0.

Proof. By Proposition 2.3, 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 G is semi-hyperbolic. By Theorem 2.6, the statement holds.

3 δ-subconformal measure

Definition 3.1. 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 3.2 ([S4]). 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 3.3 ([S4]). 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 3.4. 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) < ∞.

Remark 5. 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.

Lemma 3.5. 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 3.3, 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}.  (11)

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 (11), 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).

Lemma 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 Möbius transformation in G 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 1.34, 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.  (12)

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,  (13)

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 (12), 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,  (14)

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 (13) and (14), we get

∥(f_{i_{k}} ⋯ f_{i_{1}})′(z)∥_{V_{i}\P (G), V_{j}\P (G)} > 1, for each z ∈ U,  (15)

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 1.32, we can show that there exist 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 → ∞.  (16)

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], (16) 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 → ∞,  (17)

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 (17) 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 lemma.

By Lemma 3.5 and Lemma 3.6 , we can show the next result.

Theorem 3.7. Let G = ⟨f_{1}, f_{2}, …, f_{m}⟩ be a finitely generated rational semigroup. Assume that G is sub-hyperbolic, for each g ∈ G there is no super attracting fixed point of g in J(G), there exists an element of G with the degree at least two and each Möbius transformation in G is loxodromic. Then

dim_{H}(J(G)) ≤ s(G) ≤ s_{0}(G).

Proof. First we will show that F (G) ≠ ∅. Assume J(G) = \overline{C}. Let h ∈ G be an element with the degree at least two. Since G is sub-hyperbolic, the element h has a super attracting fixed point. But this contradicts to the assumption of our theorem. Hence F (G) ≠ ∅.

We can assume that ∞ ∈ F (G). We set E = G^{−1}(P (G) ∩ J(G)). Then E is a countable set. Fix any z_{0} ∈ J(G) \ E. By the same arguments as that in the proof of Theorem 1.32, we can show that there exist an ϵ > 0 and a sequence (g_{n}) of elements of G such that for each n, g_{n+1} = h_{n}g_{n} where h_{n} is an element of G and

g_{n}(z_{0}) ∈ J(G) \ D(P (G), ϵ).

By Lemma 3.6, we can show that

|g′_{n}(z_{0})| → ∞, as n → ∞.  (18)

For each n, let α_{n} be the analytic inverse branch of g_{n} in D(g_{n}(z_{0}), ϵ) such that α_{n}(g_{n}(z_{0})) = z_{0}. By the Koebe distortion theorem, there exist constants a > 0 and b ≥ 1 such that for each n ∈ N, if we set r_{n} = \frac{ϵ|α′_{n}(g_{n}(z_{0}))|}{a}, then

D(z_{0}, r_{n}) ⊂ α_{n}(D(g_{n}(z_{0}), ϵ)), D(z_{0}, \frac{r_{n}}{2} ) ⊃ α_{n}(D(g_{n}(z_{0}), \frac{ϵ}{b})).

Note that we can take the constants a and b independent of n and z_{0}. By (18), we have r_{n} → 0 as n → ∞. By Lemma 3.5, the statement of our theorem holds.

Theorem 3.8. 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 Möbius transformation in G is loxodromic and J(G)^{i} = ∅. Then we have

dim_{H}(J(G)) ≤ s(G) ≤ s_{0}(G).

Proof. We can assume ∞ ∈ F (G). Let x be any point of J(G). Since we have J(G) = ∪_{j=1}^{m}f_{j}^{−1}(J(G)), for each n ∈ N there exists an element g_{n} ∈ G which is in the form f_{w_{1}} ⋯ f_{w_{n}} 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 Lemma 2.5, if we take a δ smaller, then

diam (D_{g_{n}}(g_{n}(x), δ)) → 0, as n → ∞.  (19)

By Lemma 1.10, 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 Lemma 1.9, 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 (19), we have |ϕ′_{n}(0)| → 0 as n → ∞. Applying Lemma 3.5, we get

dim_{H}(J(G)) ≤ s(G).

By Theorem 3.2, we have s(G) ≤ s_{0}(G).

Example 3.9. 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. 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 2.6, we have m_{2}(J(G)) = 0, where we denote by m_{2} the 2-dimensional Lebesgue measure. By Theorem 3.7 and Proposition 3.4, we have also

dim_{H}(J(G)) ≤ s(G) ≤ s_{0}(G) ≤ 2.

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