Large deviations for return times in non-rectangle sets for axiom a diffeomorphisms
Renaud Leplaideur Benoît Saussol
Discrete & Continuous Dynamical Systems - A 2008, 22(1&2): 327-344 doi: 10.3934/dcds.2008.22.327
For Axiom A diffeomorphisms and equilibrium states, we prove a Large deviations result for the sequence of successive return times into a fixed Borel set, under some assumption on the boundary. Our result relies on and extends the work by Chazottes and Leplaideur who considered cylinder sets of a Markov partition.
keywords: thermodynamic formalism. large deviations return times
Recurrence rate in rapidly mixing dynamical systems
Benoît Saussol
Discrete & Continuous Dynamical Systems - A 2006, 15(1): 259-267 doi: 10.3934/dcds.2006.15.259
For measure preserving dynamical systems on metric spaces we study the time needed by a typical orbit to return back close to its starting point. We prove that when the decay of correlation is super-polynomial the recurrence rates and the pointwise dimensions are equal. This gives a broad class of systems for which the recurrence rate equals the Hausdorff dimension of the invariant measure.
keywords: multifractal analysis. Dimension theory Poincaré recurrences
Pointwise dimensions for Poincaré recurrences associated with maps and special flows
V. Afraimovich Jean-René Chazottes Benoît Saussol
Discrete & Continuous Dynamical Systems - A 2003, 9(2): 263-280 doi: 10.3934/dcds.2003.9.263
We introduce pointwise dimensions and spectra associated with Poincaré recurrences. These quantities are then calculated for any ergodic measure of positive entropy on a weakly specified subshift. We show that they satisfy a relation comparable to Young's formula for the Hausdorff dimension of measures invariant under surface diffeomorphisms. A key-result in establishing these formula is to prove that the Poincaré recurrence for a 'typical' cylinder is asymptotically its length. Examples are provided which show that this is not true for some systems with zero entropy. Similar results are obtained for special flows and we get a formula relating spectra for measures of the base to the ones of the flow.
keywords: special flows. pointwise dimensions spectra for measures Poincaré recurrences
An elementary way to rigorously estimate convergence to equilibrium and escape rates
Stefano Galatolo Isaia Nisoli Benoît Saussol
Journal of Computational Dynamics 2015, 2(1): 51-64 doi: 10.3934/jcd.2015.2.51
We show an elementary method to obtain (finite time and asymptotic) computer assisted explicit upper bounds on convergence to equilibrium (decay of correlations) and escape rates for systems satisfying a Lasota Yorke inequality. The bounds are deduced from the ones of suitable approximations of the system's transfer operator. We also present some rigorous experiments on some nontrivial example.
keywords: interval arithmetics Ulam method. Rigorous computation decay of correlation convergence to equilibrium
Dimensions for recurrence times: topological and dynamical properties
Vincent Penné Benoît Saussol Sandro Vaienti
Discrete & Continuous Dynamical Systems - A 1999, 5(4): 783-798 doi: 10.3934/dcds.1999.5.783
In this paper we give new properties of the dimension introduced by Afraimovich to characterize Poincaré recurrence and which we proposed to call Afraimovich-Pesin's (AP's) dimension. We will show in particular that AP's dimension is a topological invariant and that it often coincides with the asymptotic distribution of periodic points : deviations from this behavior could suggest that the AP's dimension is sensitive to some "non-typical" points.
keywords: topological invariant. Carathéodory construction dimensions Recurrence times topological entropy

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