Hiroki SUMI

### Abstract

We consider dynamics of semigroups of rational functions on 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. By the existence of an attractor, we can also show the contracting property with respect to the backward dynamics. 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 $#(\partial O\cap J(G))<\infty$, then 2-dimensional Lebesgue measure of the Julia set is equal to 0.

Next, we will consider constructing $\delta$-subconformal measures. If a rational semigroup has at most countably many elements, then we can construct $\delta$-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 $\delta .$

submission: April 21, 1998
revised to 99-1: February 16, 1999

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