Orbital shadowing, $ω$-limit sets and minimality

Joel Mitchell

doi:10.1016/j.topol.2019.106903

paper

Orbital shadowing, $ω$ -limit sets and minimality

arXiv:1909.03061 · doi:10.1016/j.topol.2019.106903

Abstract

Let $X$ be a compact Hausdorff space, with uniformity $U$ , and let $f : X \to X$ be a continuous function. For $D \in U$ , 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 $ω$ -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 al: 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.

7 pages. arXiv admin note: text overlap with arXiv:1907.02446

Orbital shadowing, ω-limit sets and minimality

Abstract

Orbital shadowing, $ω$ -limit sets and minimality