Hiroki SUMI
Skew product maps related to finitely generated rational semigroups
Abstract
We will define skew product maps related to a generator system
of a finitely generated rational semigroup. (A "rational semigroup"
is a semigroup generated by rational maps on the Riemann sphere.)
We will investigate the upper esitimate of Hausdorff dimension
of the Julia sets of finitely generated rational semigroups
applying the methods of thermodynamical formalisms to the
skew product maps.
We will define (backward) self-similar measure in the Julia sets, that
is, a kind of invariant measures whose projection to the
base space(space of one-sided infinite words) are
some Bernoulli measures.
We will show the uniform convergence of orbits of the
Perron-Frobenius operator which implies the uniqueness of the
measure. By using it, \ we will see
that the backward self-similar measures are exact.
We see the
metric entropy of backward self-similar measures with respect to the weight
$a=(a_{1},\ldots ,a_{m})$ is equal to
$$-\sum _{j=1}^{m}a_{j}\log a_{j}
+\sum _{j=1}^{m}a_{j}\log d_{j}$$
and we will
show that the topological entropy
of the skew product constructed by the generator system
$\{ f_{1},\ldots ,f_{m}\} $ is equal to
\[ \log (\Sigma _{j=1}^{m}\deg (f_{j}))\]
and there exists a unique maximal entropy measure $\tilde{\mu },\ $
which coincides with the backward self-similar measure
corresponding to the weight
$$ a_{0}:=(\frac{\deg (f_{1})}{\sum\limits _{j=1}^{m}\deg (f_{j})},\ \ldots ,\ \frac{\deg (f_{m})}{\sum\limits _{j=1}^{m}\deg (f_{j})}).$$
Hence the projection of the maximal entropy measure
of the skew product to the base space is equal to the
Bernoulli measure corresponding to the above weight $a_{0}.$
Applying this result
if $\{ f_{j}^{-1}(J(G))\} _{j=1,\ldots ,m}$
are mutually disjoint,\ then we get the following lower
estimate of Hausdorff dimension of the Julia set of
$G$,\
\[ \dim _{H}(J(G))\geq \frac{\log (\sum _{j=1}^{m}\deg (f_{j}))}{\int _{J(G)}\log (\| f'\| )~d\mu },\]
where $\mu =(\pi _{2})_{\ast }\tilde{\mu }$ and $f(x)= f_{i}(x)$ if $x\in f_{i}^{-1}(J(G)).$
submission: July 29, 1998
revision: March 6, 1999
revision: June 7, 1999
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