Orbital shadowing, ω-limit sets and minimality

Joel Mitchell

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)
56 Downloads (Pure)

Abstract

Let $X$ be a compact Hausdorff space, with uniformity $\mathscr{U}$, and let $f \colon X \to X$ be a continuous function. For $D \in \mathscr{U}$, a $D$-pseudo-orbit is a sequence $(x_i)$ for which $(f(x_i),x_{i+1}) \in D$ for all indices $i$. In this paper we show that pseudo-orbits trap $\omega$-limit sets in a neighbourhood of prescribed accuracy after a uniform time period. A consequence of this is a generalisation of a result of Pilyugin et alia: every system has the second weak shadowing property. By way of further applications we give a characterisation of minimal systems in terms of pseudo-orbits and show that every minimal system exhibits the strong orbital shadowing property.
Original languageEnglish
Article number106903
JournalTopology and its Applications
Volume268
DOIs
Publication statusPublished - 1 Dec 2019

Keywords

  • orbital shadowing
  • ω-limit set
  • minimal
  • pseudo-orbit
  • dynamical system

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