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Bowen’s formula for meromorphic functions

Published online by Cambridge University Press:  13 June 2011

KRZYSZTOF BARAŃSKI ,
BOGUSŁAWA KARPIŃSKA  and
ANNA ZDUNIK
KRZYSZTOF BARAŃSKI
Affiliation:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland (email: baranski@mimuw.edu.pl, A.Zdunik@mimuw.edu.pl)
BOGUSŁAWA KARPIŃSKA
Affiliation:
Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warszawa, Poland (email: bkarpin@mini.pw.edu.pl)
ANNA ZDUNIK
Affiliation:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland (email: baranski@mimuw.edu.pl, A.Zdunik@mimuw.edu.pl)
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Abstract

Let f be an arbitrary transcendental entire or meromorphic function in the class 𝒮 (i.e. with finitely many singularities). We show that the topological pressure P(f,t) for t>0 can be defined as the common value of the pressures P(f,t,z) for all z∈ℂ up to a set of Hausdorff dimension zero. Moreover, we prove that P(f,t) equals the supremum of the pressures of f∣X over all invariant hyperbolic subsets X of the Julia set, and we prove Bowen’s formula for f, i.e. we show that the Hausdorff dimension of the radial Julia set of f is equal to the infimum of the set of t, for which P(f,t) is non-positive. Similar results hold for (non-exceptional) transcendental entire or meromorphic functions f in the class ℬ (i.e. with a bounded set of singularities), for which the closure of the post-singular set does not contain the Julia set.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 4 , August 2012 , pp. 1165 - 1189
Copyright
Copyright © Cambridge University Press 2011

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