The equilibrium states for semigroups of rational maps

We consider the dynamics of skew product maps associated with finitely generated semigroups of rational maps on the Riemann sphere. We show that under some conditions on the dynamics and the potential function ψ , there exists a unique equilibrium state for ψ and a unique exp(P( ψ ) −  ψ )-conformal...

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Bibliographic Details
Published inMonatshefte für Mathematik Vol. 156; no. 4; pp. 371 - 390
Main Authors Sumi, Hiroki, Urbański, Mariusz
Format Journal Article
LanguageEnglish
Published Vienna Springer Vienna 01.04.2009
Subjects
Mathematics
Mathematics and Statistics
Secondary: 37A35
Skew product
Equilibrium state
Primary: 37F10
Rational semigroup
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Summary:We consider the dynamics of skew product maps associated with finitely generated semigroups of rational maps on the Riemann sphere. We show that under some conditions on the dynamics and the potential function ψ , there exists a unique equilibrium state for ψ and a unique exp(P( ψ ) −  ψ )-conformal measure, where P( ψ ) denotes the topological pressure of ψ .
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-008-0016-8