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Real analyticity of Hausdorff dimension for expanding rational semigroups

Published online by Cambridge University Press:  23 June 2009

HIROKI SUMI  and
MARIUSZ URBAŃSKI
HIROKI SUMI
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan (email: sumi@math.sci.osaka-u.ac.jp)
MARIUSZ URBAŃSKI
Affiliation:
Department of Mathematics, University of North Texas, Denton, TX 76203-1430, USA (email: urbanski@unt.edu)
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Abstract

We consider the dynamics of expanding semigroups generated by finitely many rational maps on the Riemann sphere. We show that for an analytic family of such semigroups, the Bowen parameter function is real-analytic and plurisubharmonic. Combining this with a result obtained by the first author, we show that if each semigroup of such an analytic family of expanding semigroups satisfies the open set condition, then the Hausdorff dimension of the Julia set is a real-analytic and plurisubharmonic function of the parameter. Moreover, we provide an extensive collection of examples of analytic families of semigroups satisfying all of the above conditions and we analyze in detail the corresponding Bowen’s parameters and Hausdorff dimension function.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 2 , April 2010 , pp. 601 - 633
Copyright
Copyright © Cambridge University Press 2009

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