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Shadowing, chain transitive sets with nonempty interior and attractor boundaries

Noriaki Kawaguchi

TL;DR

The paper develops a unified shadowing framework for $C^1$ dynamics to analyze chain transitive sets with nonempty interior and attractor boundaries. By establishing that shadowing, together with L-shadowing, forces strong topological constraints on invariant sets, it proves that such a set must be clopen and, in connected spaces, coincides with the entire space, yielding a mixing behavior. It also shows that attractor boundaries are chain stable under shadowing, linking local pseudo-orbit compatibility to global boundary structure. These results refine and extend known results on generic dynamics and attractor boundaries, providing a cohesive perspective via shadowing concepts. The work culminates with several corollaries and an appendix establishing a finiteness property for attractor-boundary components in locally connected spaces.

Abstract

We examine certain phenomena in $C^1$-dynamics from a viewpoint of shadowing and improve a known result on hyperbolic sets. We also review a result on the stability of attractor boundaries from the same viewpoint and derive several additional results.

Shadowing, chain transitive sets with nonempty interior and attractor boundaries

TL;DR

The paper develops a unified shadowing framework for dynamics to analyze chain transitive sets with nonempty interior and attractor boundaries. By establishing that shadowing, together with L-shadowing, forces strong topological constraints on invariant sets, it proves that such a set must be clopen and, in connected spaces, coincides with the entire space, yielding a mixing behavior. It also shows that attractor boundaries are chain stable under shadowing, linking local pseudo-orbit compatibility to global boundary structure. These results refine and extend known results on generic dynamics and attractor boundaries, providing a cohesive perspective via shadowing concepts. The work culminates with several corollaries and an appendix establishing a finiteness property for attractor-boundary components in locally connected spaces.

Abstract

We examine certain phenomena in -dynamics from a viewpoint of shadowing and improve a known result on hyperbolic sets. We also review a result on the stability of attractor boundaries from the same viewpoint and derive several additional results.
Paper Structure (6 sections, 24 theorems, 60 equations)

This paper contains 6 sections, 24 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A lemma
  4. Proof of Theorem 1.1 and corollaries
  5. Proof of Theorem 1.2 and corollaries
  6. Remarks

Key Result

Theorem 1.1

Let $f\colon X\to X$ be a homeomorphism and let $\Lambda$ be a closed $f$-invariant subset of $X$. If then $\Lambda$ is clopen in $X$; therefore, if furthermore $X$ is connected, then $X=\Lambda$ and $f$ is a mixing homeomorphism.

Theorems & Definitions (53)

  • Theorem 1.1
  • Remark 1.1
  • Corollary 1.1
  • Corollary 1.2
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 43 more