## Shadowing and internal chain transitivity

### Volume 222 / 2013

#### Abstract

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.