# Bowen Parameter and Hausdorff Dimension for Expanding Rational Semigroups - Mathematics > Dynamical Systems

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Abstract: We consider the dynamics of rational semigroups semigroups of rational mapson the Riemann sphere. We estimate the Bowen parameters zeros of the pressurefunctions and the Hausdorff dimensions of the Julia sets of expanding finitelygenerated rational semigroups. We show that the Bowen parameter is larger thanor equal to the ratio of the entropy of the skew product map $F$ and theLyapunov exponent of $F$ with respect to the maximal entropy measure for $F$.Moreover, we show that the equality holds if and only if the generators $f {j}$are simultaneously conjugate to the form $a {j}z^{\pm d}$ by a linearfractional transformation. Furthermore, we show that there are plenty ofexpanding finitely generated rational semigroups such that the Bowen parameteris strictly larger than two.

Autor: Hiroki Sumi, Mariusz Urbanski

Fuente: https://arxiv.org/