Strongly distributional chaos of irregular orbits that are not uniformly hyperbolic

TL;DR

This paper establishes a new form of strong distributional chaos in irregular orbits of certain dynamical systems, extending chaos concepts beyond uniform hyperbolicity and applying to a broad class of systems.

Contribution

It introduces a general framework for strongly distributional chaos applicable to systems with or without specification properties, including various hyperbolic and non-hyperbolic systems.

Findings

Strong distributional chaos exists in non-uniform hyperbolic irregular orbits.

Various fractal sets are shown to be strongly distributional chaotic.

The framework applies to a wide range of systems, including Anosov diffeomorphisms and $eta$-shifts.

Abstract

In this article we prove that for a diffeomorphism on a compact Riemannian manifold, if there is a nontrival homoclinic class that is not uniformly hyperbolic or the diffeomorphism is a $C^{1+\alpha}$ and there is a hyperbolic ergodic measure whose support is not uniformly hyperbolic, then we find a type of strongly distributional chaos which is stronger than usual distributional chaos and Li-Yorke chaos in the set of irregular orbits that are not uniformly hyperbolic. Meanwhile, we prove that various fractal sets are strongly distributional chaotic, such as irregular sets, level sets, several recurrent level sets of points with different recurrent frequency, and some intersections of these fractal sets. In the process of proof, we give an abstract general mechanism to study strongly distributional chaos provided that the system has a sequence of nondecreasing invariant compact subsets such that every subsystem has exponential specification property, or has exponential shadowing property and transitivity. The advantage of this abstract framework is that it is not only applicable in systems with specification property including transitive Anosov diffeomorphisms, mixing expanding maps, mixing subshifts of finite type and mixing sofic subshifts but also applicable in systems without specification property including $\beta$-shifts, Katok map, generic systems in the space of robustly transitive diffeomorphisms and generic volume-preserving diffeomorphisms.