Exceptional sets for nonuniformly hyperbolic diffeomorphisms

TL;DR

This paper investigates the size and complexity of exceptional sets in nonuniformly hyperbolic surface diffeomorphisms, showing they retain significant entropy and dimension under certain conditions.

Contribution

It establishes lower bounds on the topological entropy and Hausdorff dimension of exceptional sets relative to hyperbolic measures.

Findings

Exceptional sets have entropy at least that of the hyperbolic measure.

Hausdorff dimension of exceptional sets is at least the dimension of the measure.

Results apply when hyperbolic structures or SRB measures exist.

Abstract

For a surface diffeomorphism, a compact invariant locally maximal set $W$ and some subset $A\subset W$ we study the $A$-exceptional set, that is, the set of points whose orbits do not accumulate at $A$. We show that if the Hausdorff dimension of $A$ is smaller than the Hausdorff dimension $d$ of some ergodic hyperbolic measure, then the topological entropy of the exceptional set is at least the entropy of this measure and its Hausdorff dimension is at least $d$. Particular consequences occur when there is some a priori defined hyperbolic structure on $W$ and, for example, if there exists an SRB measure.