# American Institute of Mathematical Sciences

August  2005, 5(3): 565-586. doi: 10.3934/dcdsb.2005.5.565

## Entropy estimation and fluctuations of hitting and recurrence times for Gibbsian sources

 1 CNRS-CPHT, École Polytechnique, 91128 Palaiseau Cedex, France 2 Instituto de Física--UASLP, 78000 San Luis Potosí, SLP, Mexico

Received  January 2004 Revised  November 2004 Published  May 2005

Motivated by entropy estimation from chaotic time series, we pro- vide a comprehensive analysis of hitting times of cylinder sets in the setting of Gibbsian sources. We prove two strong approximation results from which we easily deduce pointwise convergence to entropy, lognormal fluctuations, precise large deviation estimates and an explicit formula for the hitting-time multifractal spectrum. It follows from our analysis that the hitting time of a n-cylinder fluctuates in the same way as the inverse measure of this n-cylinder at 'small scales', but in a different way at 'large scales'. In particular, the Rényi entropy differs from the hitting-time spectrum, contradicting a naive ansatz. This phenomenon was recently numerically observed for return times that are more di±cult to handle theoretically. The results we obtain for return times, though less precise than for hitting times, complete the available ones.
Citation: Jean René Chazottes, E. Ugalde. Entropy estimation and fluctuations of hitting and recurrence times for Gibbsian sources. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 565-586. doi: 10.3934/dcdsb.2005.5.565
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