Shifts of finite type as fundamental objects in the theory of shadowing

Chris Good, Jonathan Meddaugh

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)
157 Downloads (Pure)

Abstract

Shifts of finite type and the notion of shadowing, or pseudo-orbit
tracing, are powerful tools in the study of dynamical systems. In this paper we prove that there is a deep and fundamental relationship between these two concepts. Let X be a compact totally disconnected space and f : X → X a continuous map.We demonstrate that f has shadowing if and only if the system
( f, X) is (conjugate to) the inverse limit of a directed system satisfying the Mittag-Leffler condition and consisting of shifts of finite type. In particular, this implies that, in the case that X is the Cantor set, f has shadowing if and only if ( f, X)is the inverse limit of a sequence satisfying the Mittag-Leffler condition and consisting of shifts of finite type. Moreover, in the general compact metric case, where X is not necessarily totally disconnected, we prove that f has shadowing if ( f, X) is a factor of the inverse limit of a sequence satisfying the Mittag-Leffler condition and consisting of shifts of finite type by a quotient that almost lifts pseudo-orbits.
Original languageEnglish
Pages (from-to)715–736
Number of pages22
JournalInventiones Mathematicae
Volume220
DOIs
Publication statusPublished - 12 Dec 2019

Keywords

  • discrete dynamical system
  • inverse limits
  • pseudo-orbit tracing
  • shadowing
  • shifts of finite type

ASJC Scopus subject areas

  • General Mathematics

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