Asymptotic measure-expansiveness for generic diffeomorphisms

Manseob Lee

doi:10.1515/math-2021-0037

Abstract

Abstract In this paper, we will assume M M to be a compact smooth manifold and f : M → M f:M\to M to be a diffeomorphism. We herein demonstrate that a C 1 {C}^{1} generic diffeomorphism f f is Axiom A and has no cycles if f f is asymptotic measure expansive. Additionally, for a C 1 {C}^{1} generic diffeomorphism f f , if a homoclinic class H ( p , f ) H\left(\hspace{0.08em}p,f) that contains a hyperbolic periodic point p p of f f is asymptotic measure-expansive, then H ( p , f ) H\left(\hspace{0.08em}p,f) is hyperbolic of f f .

Highlights

Throughout this paper, we will assume M to be a compact smooth manifold and d to be the distance on M induced by a Riemannian metric ∥⋅∥
Expansiveness means that if any two real orbits are separated by a small distance, the two orbits are identical, and it is appropriate for studying smooth dynamic systems
Mañé [17] proved that a C1 robustly expansive diffeomorphism f is quasi Anosov, i.e., the set {∥Df n(v)∥ : n ∈ } is unbounded for all v ∈ TM⧹{0}

Summary

Introduction

Throughout this paper, we will assume M to be a compact smooth manifold and d to be the distance on M induced by a Riemannian metric ∥⋅∥. Lee [26] proved that a C1 generic measure expansive diffeomorphism f satisfies Axiom A and has no-cycle. Note that if a diffeomorphism f satisfies Axiom A, the nonwandering set Ω( f ) is a disjoint union of transitive invariant closed subsets. These sets are homoclinic classes that each contain a hyperbolic periodic point. Lee proved in [32] that a homoclinic class H(p, f ) is hyperbolic if it is measure-expansive for a C1 generic diffeomorphism f. A homoclinic class H(p, f ) is hyperbolic if it is asymptotic measure-expansive for a C1 generic f ∈ Diff(M)

Proof of Theorem A

Proof of Theorem B