Entropy of irregular points that are not uniformly hyperbolic

Abstract

In this article we prove that for a C 1+α 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.Theorem 1.1. [16, Corollary B] Let M be a compact Riemannian manifold of dimension at least 2 and f be a C 1+α diffeomorphism. Then one has h top (f, IR(f )) ≥ sup{h top (f, Λ) : Λ is a transitive locally maximal hyperbolic set} = sup{h µ (f ) : µ is a hyperbolic ergodic measure}.

High pointwise emergence via saturated sets

“…In this section, we give an abstract framework in which we show I φ ( f ) ∩ E α has positive entropy using the results of saturated sets obtained in [14,16,30].…”

“…When |V f (x)| ⩾ 2, where |A| denotes the cardinality of the set A, we say that x is a irregular point. The set of irregular points is denoted by IR and have been studied a lot, for example, see [3,6,10,14,24,29,34,37].…”

“…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]).…”