Quantitative Pesin theory for Anosov diffeomorphisms and flows

Sébastien Gouëzel (1) , Luchezar Stoyanov (2)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 UWA - The University of Western Australia

Pesin sets are measurable sets along which the behavior of a matrix cocycle above a measure preserving dynamical system is explicitly controlled. In uniformly hyper-bolic dynamics, we study how often points return to Pesin sets under suitable conditions on the cocycle: if it is locally constant, or if it admits invariant holonomies and is pinching and twisting, we show that the measure of points that do not return a linear number of times to Pesin sets is exponentially small. We discuss applications to the exponential mixing of contact Anosov flows, and counterexamples illustrating the necessity of suitable conditions on the cocycle.

Domaines: Systèmes dynamiques [math.DS]

Sébastien Gouëzel, Luchezar Stoyanov. Quantitative Pesin theory for Anosov diffeomorphisms and flows. Ergodic Theory and Dynamical Systems, 2019, 39 (1), pp.159-200. ⟨hal-01382977⟩