Homeomorphisms of the pseudo-arc are essentially small
type: Item
instance of: scholarly article
title: Homeomorphisms of the pseudo-arc are essentially small
zbMATH Open document ID: 0691.54020
DOI: 10.1016/0166-8641(90)90081-C
published in: Topology and its Applications
publication date: 1990-01-01T00:00:00Z
review text: The pseudo-arc is a nondegenerate, hereditarily indecomposable, chainable continuum. Among its many interesting properties are its homogeneity, hereditary equivalence, and the fact that though chainable it admits many effective compact group actions. It is known that every self map of a pseudo-arc is a near-homeomorphism and that every homeomorphism of the pseudo-arc is a product of \(\epsilon\)-homeomorphisms. Recently there has been increasing interest in the dynamics of the pseudo-arc, with Kennedy proving the existence of transitive homeomorphisms. Barge has asked whether there exist such homeomorphisms which are arbitrarily close to the identity. We provide a positive answer to this question and show that any dynamics occurring in homeomorphisms of the pseudo-arc occurs in homeomorphisms which are arbitrarily close to the identity. Specifically, we show that every homeomorphism of the pseudo-arc is conjugate to an \(\epsilon\)-homeomorphism, for each \(\epsilon >0\).
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54F50
zbMATH DE Number: 4131245
zbMATH Keywords: epsilon-homeomorphism
zbMATH Keywords: pseudo-arc
zbMATH Keywords: chainable continuum
author: Wayne Lewis
MaRDI profile type: Publication
cites work: A homogeneous indecomposable plane continuum
cites work: A Transitive Homeomorphism on the Pseudoarc which is Semiconjugate to the Tent Map
cites work: Q3697892
cites work: Most Maps of the Pseudo-Arc are Homeomorphisms
cites work: An Indecomposable Plane Continuum Which is Homeomorphic to Each of Its Nondegenerate Subcontinua
full work available at URL: 0166-8641(90)90081-c
OpenAlex ID: W2087175875
description: scientific article; zbMATH DE number 4131245