# American Institute of Mathematical Sciences

November  2014, 34(11): 4765-4780. doi: 10.3934/dcds.2014.34.4765

## The structure of limit sets for $\mathbb{Z}^d$ actions

 1 Department of Mathematics, Baylor University, Waco, TX 76798-7328, United States, United States

Received  February 2013 Revised  February 2014 Published  May 2014

Central to the study of $\mathbb{Z}$ actions on compact metric spaces is the $\omega$-limit set, the set of all limit points of a forward orbit. A closed set $K$ is internally chain transitive provided for every $x,y\in K$ there is an $\epsilon$-pseudo-orbit of points from $K$ that starts with $x$ and ends with $y$. It is known in several settings that the property of internal chain transitivity characterizes $\omega$-limit sets. In this paper, we consider actions of $\mathbb{Z}^d$ on compact metric spaces. We give a general definition for shadowing and limit sets in this setting. We characterize limit sets in terms of a more general internal property which we call internal mesh transitivity.
Citation: Jonathan Meddaugh, Brian E. Raines. The structure of limit sets for $\mathbb{Z}^d$ actions. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4765-4780. doi: 10.3934/dcds.2014.34.4765
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