Special Ergodic Theorem for hyperbolic maps

TL;DR

This paper proves that for hyperbolic maps, the set of points with atypical Birkhoff averages has Hausdorff dimension less than the manifold's dimension, extending the special ergodic theorem to various classes of dynamical systems.

Contribution

It establishes the special ergodic theorem for hyperbolic maps using large deviations principles, covering transitive hyperbolic attractors and other classes.

Findings

The special ergodic theorem holds for transitive hyperbolic attractors of C^2-diffeomorphisms.

The theorem applies to some partially hyperbolic non-uniformly expanding maps.

Hausdorff dimension of atypical points set is less than the manifold's dimension.

Abstract

Let f be a self-map of a compact manifold M, admitting an global SRB measure \mu. For a continuous test function \phi on M and a constant \alpha>0, consider the set of the initial points for which the Birkhoff time averages of the function \phi differ from its \mu--space average by at least \alpha. As the measure \mu is an SRB one, the intersection of this set with the basin of attraction of \mu should have zero Lebesgue measure. The \emph{special ergodic theorem}, whenever it holds, claims that, moreover, this intersection has the Hausdorff dimension less than the dimension of M. We prove that for Lipschitz maps, the special ergodic theorem follows from the dynamical large deviations principle. Applying theorems of L. S. Young and of V. Araujo and M. J. Pacifico, we conclude that the special ergodic theorem holds for transitive hyperbolic attractors of C^2-diffeomorphisms, as well as for some other known classes of maps (including the one of partially hyperbolic non-uniformly expanding maps).