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Local Shadowing and Entropy for Homeomorphisms

Piotr Oprocha, Elias Rego

TL;DR

The paper develops a local shadowing framework for homeomorphisms by introducing positively shadowable points and measures and establishing how local shadowing generates semi-horseshoes that encode entropy-rich dynamics. Through a sequence of constructions, it proves that invariant measures supported on non-equicontinuous chain-recurrent classes can be approximated by ergodic measures with controlled entropy, yielding entropy-flexibility results both locally and globally, especially under entropy-expansiveness and shadowing. It also provides explicit mechanisms to build semi-horseshoes from pointwise data, clarifies the relation between local versus global shadowing, and extends the theory to non-invertible maps. The work thus links pointwise shadowing properties with measure-theoretic entropy, enabling precise entropy control and revealing how local dynamical features shape global chaotic behavior.

Abstract

In this work, we investigate the dynamics of homeomorphisms through the lens of the local shadowing theory. We study the influence of positively shadowable points and positively shadowable measures into the local entropy theory of homeomrphisms. Specifically, we use pointwise shadowing to approximate invariant measures by ergodic measures with bigger entropy and supported on semi-horseshoes, giving certain flexibility of entropy to systems with local shadowing. We further apply our findings to show that by merging some pointwise dynamical properties, one can lead to positive entropy, significantly enhancing several related results previously established in the field. Besides, we introduce new examples of dynamical systems with local shadowing phenomena, by providing a way of construct examples, but lacking of several properties present in systems with the global shadowing, showing that the local and global shadowing are, in fact, distinct phenomena.

Local Shadowing and Entropy for Homeomorphisms

TL;DR

The paper develops a local shadowing framework for homeomorphisms by introducing positively shadowable points and measures and establishing how local shadowing generates semi-horseshoes that encode entropy-rich dynamics. Through a sequence of constructions, it proves that invariant measures supported on non-equicontinuous chain-recurrent classes can be approximated by ergodic measures with controlled entropy, yielding entropy-flexibility results both locally and globally, especially under entropy-expansiveness and shadowing. It also provides explicit mechanisms to build semi-horseshoes from pointwise data, clarifies the relation between local versus global shadowing, and extends the theory to non-invertible maps. The work thus links pointwise shadowing properties with measure-theoretic entropy, enabling precise entropy control and revealing how local dynamical features shape global chaotic behavior.

Abstract

In this work, we investigate the dynamics of homeomorphisms through the lens of the local shadowing theory. We study the influence of positively shadowable points and positively shadowable measures into the local entropy theory of homeomrphisms. Specifically, we use pointwise shadowing to approximate invariant measures by ergodic measures with bigger entropy and supported on semi-horseshoes, giving certain flexibility of entropy to systems with local shadowing. We further apply our findings to show that by merging some pointwise dynamical properties, one can lead to positive entropy, significantly enhancing several related results previously established in the field. Besides, we introduce new examples of dynamical systems with local shadowing phenomena, by providing a way of construct examples, but lacking of several properties present in systems with the global shadowing, showing that the local and global shadowing are, in fact, distinct phenomena.

Paper Structure

This paper contains 13 sections, 23 theorems, 117 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Topological Dynamics
  4. Pseudo-Orbits, Chain-Recurrence and Shadowing
  5. Topological Entropy
  6. Symbolic Dynamics and Semi-Horseshoes
  7. Ergodic Theory and Metric Entropy
  8. Positively Shadowable Points
  9. Invariant Shadowable Measures
  10. Local Shadowing and Semi-Horseshoes
  11. Measure Approximation and Entropy Flexibility
  12. Positive entropy through pointwise and measurable dynamics
  13. The Non-Invertible Case

Key Result

Theorem A

Let $X$ be the Cantor set and $T\colon X\to X$ a homeomorphism. Then, there is a compact metric space $M$ containing $X$ and a homeomorphism $f: M\to M$ such that $f|_{X}=T$ and $Sh^+(f)=X$. In addition, if $Y\subset M$ is a compact and $f$-invariant set so that $f|_Y$ has the shadowing property,

Figures (1)

  • Figure 1: A homeomorphism exhibiting positively shadowable points that are not shadowable.

Theorems & Definitions (58)

  • Theorem A
  • Example 1.1
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4: LiOp
  • Lemma 2.5: LiOp
  • Lemma 2.6: LiOp
  • ...and 48 more