Hiroki SUMI
Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups
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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