Entropy of irregular points that are not uniformly hyperbolic

TL;DR

This paper establishes a lower bound for the topological entropy of irregular points in certain dynamical systems, linking it to the metric entropy of hyperbolic measures, and introduces a general method for analyzing entropy of irregular points.

Contribution

It proves a new lower bound for the entropy of irregular points in non-uniformly hyperbolic systems and presents a general framework for studying entropy via shadowing and transitivity.

Findings

Topological entropy of irregular points is at least the metric entropy of a hyperbolic ergodic measure.

Introduces an abstract mechanism for analyzing entropy of irregular points in systems with shadowing and transitivity.

Establishes a connection between non-uniform hyperbolicity and entropy of irregular points.

Abstract

In this article we prove that for a $C^{1+\alpha}$ diffeomorphism on a compact Riemannian manifold, if there is a hyperbolic ergodic measure whose support is not uniformly hyperbolic, then the topological entropy of the set of irregular points that are not uniformly hyperbolic is larger than or equal to the metric entropy of the hyperbolic ergodic measure. In the process of proof, we give an abstract general mechanism to study topological entropy of irregular points provided that the system has a sequence of nondecreasing invariant compact subsets such that every subsystem has shadowing property and is transitive.