Shadowing and internal chain transitivity

Jonathan Meddaugh ¹   ; Brian E. Raines ¹

¹ Department of Mathematics Baylor University Waco, TX 76798-7328, U.S.A.

Fundamenta Mathematicae, Tome 222 (2013) no. 3, pp. 279-287

Résumé

The main result of this paper is that a map $f:X\to X$ which has shadowing and for which the space of $\omega $-limits sets is closed in the Hausdorff topology has the property that a set $A\subseteq X$ is an $\omega $-limit set if and only if it is closed and internally chain transitive. Moreover, a map which has the property that every closed internally chain transitive set is an $\omega $-limit set must also have the property that the space of $\omega $-limit sets is closed. As consequences of this result, we show that interval maps with shadowing have the property that every internally chain transitive set is an $\omega $-limit set of a point, and we also show that topologically hyperbolic maps and certain quadratic Julia sets have a closed space of $\omega $-limit sets.

Détail

DOI : 10.4064/fm222-3-4

Keywords: main result paper map which has shadowing which space omega limits sets closed hausdorff topology has property set subseteq omega limit set only closed internally chain transitive moreover map which has property every closed internally chain transitive set omega limit set have property space omega limit sets closed consequences result interval maps shadowing have property every internally chain transitive set omega limit set point topologically hyperbolic maps certain quadratic julia sets have closed space omega limit sets