Example of a diffeomorphism for which the special ergodic theorem doesn't hold

TL;DR

This paper constructs a smooth diffeomorphism on a 4-manifold with a global SRB measure where the special ergodic theorem fails, showing that certain sets of initial points can have full Hausdorff dimension despite zero Lebesgue measure.

Contribution

It provides a counterexample demonstrating the failure of the special ergodic theorem in a smooth dynamical system with a global SRB measure.

Findings

Existence of a smooth diffeomorphism with a global SRB measure where the theorem fails.

The set of points with atypical Birkhoff averages has full Hausdorff dimension.

The set of initial points with deviation has zero Lebesgue measure.

Abstract

In this work we present an example of C^\infty-diffeomorphism of a compact 4-manifold such that it admits a global SRB measure \mu but for which the special ergodic theorem doesn't hold. Namely, for this transformation there exist a continuous function \phi and a positive constant \alpha such that the following holds: 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 has zero Lebesgue measure but full Hausdorff dimension.