Expansivity and unique shadowing

Research output: Contribution to journalArticlepeer-review

Authors

Colleges, School and Institutes

External organisations

  • School of Mathematics
  • Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria
  • Department of Mathematics
  • Baylor University
  • University of Manchester

Abstract

Let f: X → X be a continuous function on a compact metric space. We show that shadowing is equivalent to backwards shadowing and two-sided shadowing when the map f is onto. Using this we go on to show that, for expansive surjective maps the properties shadowing, two-sided shadowing, s-limit shadowing, and two-sided s-limit shadowing are equivalent. We show that f is positively expansive and has shadowing if and only if it has unique shadowing (i.e., each pseudo-orbit is shadowed by a unique point), extending a result implicit in Walter's proof that positively expansive maps with shadowing are topologically stable. We use the aforementioned result on two-sided shadowing to find an equivalent characterisation of shadowing and expansivity and extend these results to the notion of n-expansivity due to Morales.

Bibliographic note

Publisher Copyright: © 2020 American Mathematical Society Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

Details

Original languageEnglish
Pages (from-to)671-685
Number of pages15
JournalProceedings of the American Mathematical Society
Volume149
Issue number2
Publication statusE-pub ahead of print - 25 Nov 2020

Keywords

  • Expansive, Pseudo-orbit, S-limit shadowing, Shadowing

ASJC Scopus subject areas