Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homoclinic classes with shadowing property

Manseob Lee

TL;DR

This paper addresses when an isolated homoclinic class has the shadowing property under $C^1$-generic diffeomorphisms. It employs Mañé-type perturbation techniques, along with residual genericity and semi-continuity arguments, to show that if an isolated homoclinic class $H_f(p)$ is shadowable then it must be hyperbolic, specifically a hyperbolic basic set. The result establishes a precise equivalence between shadowability and hyperbolic basic structure in the generic setting, contributing to the broader conjecture that shadowing characterizes hyperbolicity. It also parallels findings for locally maximal chain transitive sets and expansiveness, reinforcing the connection between shadowing and hyperbolicity in the tame (finite) case of dynamical systems.

Abstract

We show that for $C^1$ generic diffeomorphisms, an isolated homoclinic class is shadowable if and only if homoclinic class is hyperbolic basic set.

Homoclinic classes with shadowing property

TL;DR

This paper addresses when an isolated homoclinic class has the shadowing property under -generic diffeomorphisms. It employs Mañé-type perturbation techniques, along with residual genericity and semi-continuity arguments, to show that if an isolated homoclinic class is shadowable then it must be hyperbolic, specifically a hyperbolic basic set. The result establishes a precise equivalence between shadowability and hyperbolic basic structure in the generic setting, contributing to the broader conjecture that shadowing characterizes hyperbolicity. It also parallels findings for locally maximal chain transitive sets and expansiveness, reinforcing the connection between shadowing and hyperbolicity in the tame (finite) case of dynamical systems.

Abstract

We show that for generic diffeomorphisms, an isolated homoclinic class is shadowable if and only if homoclinic class is hyperbolic basic set.

Paper Structure

This paper contains 2 sections, 10 theorems, 14 equations.

Table of Contents

  1. Introduction.
  2. Proof of Theorem \ref{['thm0']}

Key Result

Theorem 1.1

For $C^1$ generic $f,$ if $f$ is tame, then the following two conditions are equivalent:

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 7 more