Hiroki SUMI
Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and
skew products
Abstract
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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