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August  2008, 21(3): 729-747. doi: 10.3934/dcds.2008.21.729

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, Brazil
2. 

PHYMAT, Université de Toulon et du Var, Centre de Physique Théorique, CNRS, Luminy Case 907, F-13288 Marseille Cedex 9, France

Received  May 2007 Revised  October 2007 Published  April 2008

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.
Keywords: rare event, large deviations., overlapping, recurrence, Mixing, short correlation.
Mathematics Subject Classification: Primary: 60F05; Secondary: 60G10, 60G55, 37A5.
Citation: Miguel Abadi, Sandro Vaienti. Large deviations for short recurrence. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 729-747. doi: 10.3934/dcds.2008.21.729
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