Fluctuations of the nth return time for Axiom A diffeomorphisms
Jean-René Chazottes Renaud Leplaideur
Discrete & Continuous Dynamical Systems - A 2005, 13(2): 399-411 doi: 10.3934/dcds.2005.13.399
We study the time of $n$th return of orbits to some given (union of) rectangle(s) of a Markov partition for an Axiom A diffeomorphism. Namely, we prove the existence of a scaled generating function for these returns with respect to any Gibbs measure. As a by-product, we derive precise large deviation estimates and a central limit theorem for these return times. We emphasize that we look at the limiting behavior in term of number of visits (the size of the visited set is kept fixed). Our approach relies on the spectral properties of a one-parameter family of induced transfer operators on unstable leaves crossing the visited set.
keywords: large deviations Successive return times Axiom A central limit theorem. transfer operator
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

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