Expansivity and unique shadowing

Research output: Contribution to journalArticle

Authors

Colleges, School and Institutes

External organisations

  • Baylor University
  • UNAM
  • 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 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

Final Version of Record not yet available as of 20/10/2020

Details

Original languageEnglish
JournalProceedings of the American Mathematical Society
Publication statusAccepted/In press - 27 May 2020

Keywords

  • Expansive, shadowing, pseudo-orbit, s-limit shadowing