Abstract

In this article, we show that there are homeomorphisms of plane continua whose conjugacy class is residual and have the shadowing property.

Highlights

  • Let ( X, dist) be a compact metric space and denote by H( X ) the space of homeomorphisms f : X → X with the C0 distance distC0 ( f, g) = sup{dist( f ( x ), g( x )), dist( f −1 ( x ), g−1 ( x )) : x ∈ X }.A property is said to be generic if it holds on a residual subset of H( X )

  • In this article, we show that there are homeomorphisms of plane continua whose conjugacy class is residual and have the shadowing property

  • Recall that a set is Gδ if it is a countable intersection of open sets and it is residual if it contains a dense Gδ subset

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Summary

Introduction

We say that f , g ∈ H( X ) are conjugate if there is h ∈ H( X ) such that f ◦ h = h ◦ g and the conjugacy class of f is the set of all the homeomorphisms conjugate to f This result gives rise to a natural question: besides the Cantor set, which compact metric spaces have a Gδ dense conjugacy class?. In Theorem 2, we show that there are one-dimensional plane continua with a Gδ dense conjugacy class whose members have the shadowing property The proof of this result is based on Theorem 1, where we show that for a compact interval I there is a Gδ conjugacy class in H( I ) which is dense in the subset of orientation preserving homeomorphisms of I. The following open question has an affirmative answer in the examples known by the authors: if a homeomorphism has a Gδ dense conjugacy class, does it have the shadowing property?

Generic Dynamics on a Closed Segment
Genericity on a Plane One-Dimensional Continuum
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