Hiroki SUMI

Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products


A "rational semigroup" is a semigroup generated by non-constant rational maps on the Riemann sphere. We will define sub-hyperbolic and semi-hyperbolic 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.

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. The results in this section 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. We will show that any sub-hyperbolic semigroup without any superattracting fixed point of any element of the semigroup in the Julia set is semi-hyperbolic.

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 $ \sharp (\partial O\cap J(G))<\infty.$
In section 5,\ we will consider constructing $\delta $-subconformal measures. If a rational semigroup has at most countably many elements and the $\delta $-Poincar\'{e} series converges,\ then we can construct $\delta $-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 $\delta .$ To show those results, the contracting property of backward dynamics will be used.

submission: February 16, 1999
revision: June 7, 1999

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