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

Jonathan Meddaugh; Brian E. Raines

doi:10.3934/dcds.2014.34.4765

Article Contents

2014, Volume 34, Issue 11: 4765-4780. Doi: 10.3934/dcds.2014.34.4765

This issue Previous Article Non-normal numbers in dynamical systems fulfilling the specification property Next Article Self-intersections of trajectories of the Lorentz process

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

Jonathan Meddaugh^1, and
Brian E. Raines^1,

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

Received: January 31, 2013

Revised: January 31, 2014

Published: October 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 37B50, 37B10, 37B20; Secondary: 54H20.

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