# 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 - A, 2014, 34 (11) : 4765-4780. doi: 10.3934/dcds.2014.34.4765
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##### References:
 [1] F. Balibrea and C. La Paz, A characterization of the $\omega$-limit sets of interval maps,, Acta Math. Hungar., 88 (2000), 291.  doi: 10.1023/A:1026775906693.  Google Scholar [2] A. D. Barwell, C. Good, R. Knight and B. E. Raines, A characterization of $\omega$-limit sets in shift spaces,, Ergodic Theory Dynam. Systems, 30 (2010), 21.  doi: 10.1017/S0143385708001089.  Google Scholar [3] A. D. Barwell, A characterization of $\omega$-limit sets of piecewise monotone maps of the interval,, Fund. Math., 207 (2010), 161.  doi: 10.4064/fm207-2-4.  Google Scholar [4] A. D. Barwell, C. Good, P. Oprocha and B. E. Raines, Characterizations of $\omega$-limit sets of topologically hyperbolic spaces,, Discrete Contin. Dyn. Syst., 33 (2013), 1819.  doi: 10.3934/dcds.2013.33.1819.  Google Scholar [5] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics,, American Mathematical Society, (1955).   Google Scholar [6] M. W. Hirsch, H.L. Smith and X. Q. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems,, J. Dynam. Differential Equations, 13 (2001), 107.  doi: 10.1023/A:1009044515567.  Google Scholar [7] M. Hochman, On the dynamics and recursive properties of multidimensional symbolic systems,, Invent. Math., 176 (2009), 131.  doi: 10.1007/s00222-008-0161-7.  Google Scholar [8] M. Hochman and T. Meyerovitch, A characterization of the entropies of multidimensional shifts of finite type,, Ann. of Math. (2), 171 (2010), 2011.  doi: 10.4007/annals.2010.171.2011.  Google Scholar [9] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Cambridge University Press, (1995).   Google Scholar [10] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511626302.  Google Scholar [11] P. Oprocha, Chain recurrence in multidimensional time discrete dynamical systems,, Discrete Contin. Dyn. Syst., 20 (2008), 1039.  doi: 10.3934/dcds.2008.20.1039.  Google Scholar [12] P. Oprocha, Shadowing in multi-dimensional shift spaces,, Colloq. Math., 110 (2008), 451.  doi: 10.4064/cm110-2-8.  Google Scholar [13] P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982).   Google Scholar
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