Bowen parameter and Hausdorff dimension for expanding rational semigroups

Hiroki Sumi; Mariusz Urbański

doi:10.3934/dcds.2012.32.2591

Article Contents

2012, Volume 32, Issue 7: 2591-2606. Doi: 10.3934/dcds.2012.32.2591

This issue Previous Article Density of repelling fixed points in the Julia set of a rational or entire semigroup, II Next Article

Bowen parameter and Hausdorff dimension for expanding rational semigroups

Hiroki Sumi^1, and
Mariusz Urbański^2,

1.
Department of Mathematics, Graduate School of Science, Osaka University, 1-1, Machikaneyama, Toyonaka, Osaka, 560-0043
2.
Department of Mathematics, University of North Texas, Denton, TX 76203-1430

Received: November 2009

Revised: August 2010

Published: July 2012

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Abstract

We estimate the Bowen parameters 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 $\tilde{f}$ and the Lyapunov exponent of $\tilde{f}$ with respect to the maximal entropy measure for $\tilde{f}$. Moreover, we show that the equality holds if and only if the generators are simultaneously conjugate to the form $a_{j}z^{\pm d}$ by a M\"{o}bius transformation. Furthermore, we show that there are plenty of expanding finitely generated rational semigroups such that the Bowen parameter is strictly larger than $2$.

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Mathematics Subject Classification: Primary: 37F35; Secondary: 37F15.

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