Exceptional sets for average conformal dynamical systems

TL;DR

This paper investigates the size and complexity of exceptional sets in average conformal dynamical systems, focusing on their entropy and Hausdorff dimension, especially for sets with small entropy or dimension.

Contribution

It extends the understanding of exceptional sets in average conformal systems by analyzing their topological entropy and Hausdorff dimension under specific conditions.

Findings

Exceptional sets have small entropy and Hausdorff dimension under certain conditions.

The paper provides bounds for the topological entropy of exceptional sets.

It characterizes the Hausdorff dimension of limit exceptional sets in average conformal systems.

Abstract

Let $f: M \to M$ be a $C^{1+\alpha}$ map/diffeomorphism of a compact Riemannian manifold $M$ and $\mu$ be an expanding/hyperbolic ergodic $f$-invariant Borel probability measure on $M$. Assume $f$ is average conformal expanding/hyperbolic on the support set $W$ of $\mu$ and $W$ is locally maximal. For any subset $A\subset W$ with small entropy or dimension, we investigate the topological entropy and Hausdorff dimensions of the $A$-exceptional set and the limit $A$-exceptional set.