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The Rényi entropy function and the large deviation of short return times

Published online by Cambridge University Press:  21 May 2009

NICOLAI HAYDN  and
SANDRO VAIENTI
NICOLAI HAYDN
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, USA (email: nhaydn@math.usc.edu)
SANDRO VAIENTI
Affiliation:
Centre de Physique Théorique, UMR 6207, CNRS, Luminy Case 907, F-13288 Marseille Cedex 9, France Universities of Aix-Marseille I, II and Toulon-Var, Fédération de Recherche des Unités de Mathématiques de Marseille, France (email: vaienti@cpt.univ-mrs.fr)
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Abstract

We consider the Rényi entropy function for weakly ψ-mixing systems. The first main result of the paper establishes existence and regularity properties. The second main result is obtaining the decay rate for the large deviation of the return time to cylinder sets; we show that it is exponential with a rate given by the Rényi entropy function. Finally, we also obtain bounds for the free energy.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 1 , February 2010 , pp. 159 - 179
Copyright
Copyright © Cambridge University Press 2009

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