Hiroki SUMI

Skew product maps related to finitely generated rational semigroups


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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