Recurrence rate in rapidly mixing dynamical systems
Benoît Saussol
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
Large deviations for return times in non-rectangle sets for axiom a diffeomorphisms
Renaud Leplaideur Benoît Saussol
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
An elementary way to rigorously estimate convergence to equilibrium and escape rates
Stefano Galatolo Isaia Nisoli Benoît Saussol
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
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
Pointwise dimensions for Poincaré recurrences associated with maps and special flows
V. Afraimovich Jean-René Chazottes Benoît Saussol
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
Linear response in the intermittent family: Differentiation in a weighted $C^0$-norm
Wael Bahsoun Benoît Saussol
We provide a general framework to study differentiability of SRB measures for one dimensional non-uniformly expanding maps. Our technique is based on inducing the non-uniformly expanding system to a uniformly expanding one, and on showing how the linear response formula of the non-uniformly expanding system is inherited from the linear response formula of the induced one. We apply this general technique to interval maps with a neutral fixed point (Pomeau-Manneville maps) to prove differentiability of the corresponding SRB measure. Our work covers systems that admit a finite SRB measure and it also covers systems that admit an infinite SRB measure. In particular, we obtain a linear response formula for both finite and infinite SRB measures. To the best of our knowledge, this is the first work that contains a linear response result for infinite measure preserving systems.
keywords: intermittent maps. Linear response

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