Exceptional sets for nonuniformly expanding maps

TL;DR

This paper investigates the size and complexity of exceptional sets in nonuniformly expanding maps, showing they can have full entropy or large Hausdorff dimension under certain conditions, with applications to rational and multimodal maps.

Contribution

It establishes new results on the entropy and Hausdorff dimension of $A$-exceptional sets in nonuniformly expanding systems, extending classical theory to broader classes of maps.

Findings

$A$-exceptional sets can have full topological entropy when $A$'s entropy is less than the system's.

The Hausdorff dimension of $A$-exceptional sets can be at least the system's dynamical dimension.

Equality of Hausdorff and dynamical dimensions occurs when these dimensions coincide for the system.

Abstract

Given a rational map of the Riemann sphere and a subset $A$ of its Julia set, we study the $A$-exceptional set, that is, the set of points whose orbit does not accumulate at $A$. We prove that if the topological entropy of $A$ is less than the topological entropy of the full system then the $A$-exceptional set has full topological entropy. Furthermore, if the Hausdorff dimension of $A$ is smaller than the dynamical dimension of the system then the Hausdorff dimension of the $A$-exceptional set is larger than or equal to the dynamical dimension, with equality in the particular case when the dynamical dimension and the Hausdorff dimension coincide. We discuss also the case of a general conformal $C^{1+\alpha}$ dynamical system and, in particular, certain multimodal interval maps on their Julia set.