Strongly distributional chaos of irregular orbits that are not uniformly hyperbolic

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+α 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 β-shifts, Katok map, generic systems in the space of robustly transitive diffeomorphisms and generic volume-preserving diffeomorphisms.

Entropy of irregular points that are not uniformly hyperbolic

“…And in the present paper, for possibly more applications we give an abstract general mechanism to study irregular points provided that the system has a sequence of nondecreasing invariant compact subsets such that every subsystem has shadowing property and is transitive. In our previous paper [20], we proved that the irregular points that are not uniformly hyperbolic are strongly distributional chaos. In this paper, we will prove that the irregular points that are not uniformly hyperbolic have strong dynamical complexity in the sense of topological entropy.…”

“…[20, Proposition 4.9] Suppose that (X, f ) is a dynamical system. Let Y ⊆ X be a non-empty compact f -invariant set.…”

Strongly distributional chaos in the sets of twelve different types of non-recurrent points

“…Readers can refer to [17,55,58] for the definition of DC2 and DC3 if necessary. Now we recall from [27] a kind of chaos, strongly distributional chaos, which is stronger than usual distributional chaos and Li-Yorke chaos. For any positive integer n, points x, y ∈ M and…”

“…For every β 1 > 1, there exists 1 < β 2 < β 1 such that and (Σ β 2 , σ) satisfies the shadowing property. By [27,Proposition 6.4] every subshift with shadowing property satisfies exponential shadowing property. So (Σ β 2 , σ) has exponential shadowing property.…”

High pointwise emergence via saturated sets

“…This framework is applicable to uniformly hyperbolic systems, but it is not applicable to general homoclinic classes and diffeomorphisms preserving hyperbolic ergodic measures. Recently, a new framework in [13,14] developed to study dynamical complexity of saturated set of such systems enables us to consider dynamical complexity of E α . Also, Strongly distributional chaos (definition 2.10) was introduced in [13], it is stronger than distributional chaos of type 1 ([13, proposition 2.5]).…”