Bowen Parameter and Hausdorff Dimension for Expanding Rational Semigroups

TL;DR

This paper investigates the relationship between Bowen parameters and Hausdorff dimensions of Julia sets for expanding rational semigroups, providing bounds and conditions for equality, and exploring cases with Bowen parameters exceeding two.

Contribution

It offers new estimates for Bowen parameters and Hausdorff dimensions in the context of expanding rational semigroups, including characterizations of equality cases and examples with large Bowen parameters.

Findings

Bowen parameter is at least the ratio of entropy to Lyapunov exponent.

Equality in the Bowen parameter occurs only for specific conjugate forms of generators.

Existence of expanding rational semigroups with Bowen parameter greater than two.

Abstract

We consider the dynamics of rational semigroups (semigroups of rational maps) on the Riemann sphere. We estimate the Bowen parameters (zeros of the pressure functions) and the Hausdorff dimensions of the Julia sets of expanding finitely generated rational semigroups. We show that the Bowen parameter is larger than or equal to the ratio of the entropy of the skew product map $F$ and the Lyapunov 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 linear fractional transformation. Furthermore, we show that there are plenty of expanding finitely generated rational semigroups such that the Bowen parameter is strictly larger than two.