The main result of this paper is that a map f: X → X which has shadowing and for which the space of ω-limits sets is closed in the Hausdorff topology has the property that a set A ⊆ X is an ω-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 ω-limit set must also have the property that the space of ω-limit sets is closed. As consequences of this result, we show that interval maps with...

We study shadowing properties of continuous actions of the groups ${\mathbb{Z}}^p$ and ${\mathbb{Z}}^p\times {ℝ}^p$. Necessary and sufficient conditions are given under which a linear action of ${\mathbb{Z}}^p$ on ${ℂ}^m$ has a Lipschitz shadowing property.

We show that the class of expansive ${\mathbb{Z}}^d$ actions with P.O.T.P. is wider than the class of actions topologically hyperbolic in some direction $\nu \in {\mathbb{Z}}^d$. Our main tool is an extension of a result by Walters to the multi-dimensional symbolic dynamics case.