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A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups

Published online by Cambridge University Press:  02 February 2004

MARC KESSEBÖHMER  and
BERND O. STRATMANN
MARC KESSEBÖHMER
Affiliation:
Universität Bremen, Fachbereich für Mathematik und Informatik, Bibliothekstraße 1, D–28359 Bremen, Germany (e-mail: [email protected])
BERND O. STRATMANN
Affiliation:
University of St. Andrews, School of Mathematics and Statistics, North Haugh, Fife KY16 9SS, UK (e-mail: [email protected])
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Abstract

We elaborate thermodynamic and multifractal formalisms for general classes of potential functions and their average growth rates. We then apply these formalisms to certain geometrically finite Kleinian groups which may have parabolic elements of different ranks. We show that for these groups our revised formalisms give access to a description of the spectrum of ‘homological growth rates’ in terms of Hausdorff dimension. Furthermore, we derive necessary and sufficient conditions for the existence of ‘thermodynamic phase transitions’.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 24 , Issue 1 , February 2004 , pp. 141 - 170
Copyright
2004 Cambridge University Press

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