Large deviations for short recurrence

Miguel Abadi; Sandro Vaienti

doi:10.3934/dcds.2008.21.729

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2008, Volume 21, Issue 3: 729-747. Doi: 10.3934/dcds.2008.21.729

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Large deviations for short recurrence

Miguel Abadi^1, and
Sandro Vaienti^2,

1.
Instituto de Matemática, Estatística e Computaçâo Científica, Universidade Estadual de Campinas, Pça Sergio B. de Holanda 651, Cid. Univ. Campinas, SP
2.
PHYMAT, Université de Toulon et du Var, Centre de Physique Théorique, CNRS, Luminy Case 907, F-13288 Marseille Cedex 9

Received: April 30, 2007

Revised: September 30, 2007

Published: August 2008

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Abstract

Over a $\psi$-mixing dynamical system we consider the function $\tau(C_n)$ $/n$ in the limit of large $n$, where $\tau(C_n)$ is the first return of a cylinder of length $n$ to itself. Saussol et al. ([30]) proved that this function is constant almost everywhere if the $C_n$ are chosen in a descending sequence of cylinders around a given point. We prove upper and lower general bounds for its large deviation function. Under mild assumptions we compute the large deviation function directly and show that the limit corresponds to the Rényi's entropy of the system. We finally compute the free energy function of $\tau(C_n)$ $/n$. We illustrate our results with a few examples.

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Mathematics Subject Classification: Primary: 60F05; Secondary: 60G10, 60G55, 37A50.

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