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

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

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, United States, United States

Received February 2013 Revised February 2014 Published May 2014

Abstract
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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.

Keywords: weak incompressibility, Omega-limit set, $\omega$-limit set, internal chain transitivity., pseudo-orbit tracing property, shadowing.

Mathematics Subject Classification: Primary: 37B50, 37B10, 37B20; Secondary: 54H2.

Citation: Jonathan Meddaugh, Brian E. Raines. The structure of limit sets for $\mathbb{Z}^d$ actions. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4765-4780. doi: 10.3934/dcds.2014.34.4765

References:

[1]	F. Balibrea and C. La Paz, A characterization of the $\omega$-limit sets of interval maps, Acta Math. Hungar., 88 (2000), 291-300. doi: 10.1023/A:1026775906693.
[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), 21-31. doi: 10.1017/S0143385708001089.
[3]	A. D. Barwell, A characterization of $\omega$-limit sets of piecewise monotone maps of the interval, Fund. Math., 207 (2010), 161-174. doi: 10.4064/fm207-2-4.
[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), 1819-1833. doi: 10.3934/dcds.2013.33.1819.
[5]	W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, American Mathematical Society, Providence, R. I., 1955.
[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), 107-131. doi: 10.1023/A:1009044515567.
[7]	M. Hochman, On the dynamics and recursive properties of multidimensional symbolic systems, Invent. Math., 176 (2009), 131-167. doi: 10.1007/s00222-008-0161-7.
[8]	M. Hochman and T. Meyerovitch, A characterization of the entropies of multidimensional shifts of finite type, Ann. of Math. (2), 171 (2010), 2011-2038. doi: 10.4007/annals.2010.171.2011.
[9]	A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995.
[10]	D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302.
[11]	P. Oprocha, Chain recurrence in multidimensional time discrete dynamical systems, Discrete Contin. Dyn. Syst., 20 (2008), 1039-1056. doi: 10.3934/dcds.2008.20.1039.
[12]	P. Oprocha, Shadowing in multi-dimensional shift spaces, Colloq. Math., 110 (2008), 451-460. doi: 10.4064/cm110-2-8.
[13]	P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.

show all references

References:

[1]	F. Balibrea and C. La Paz, A characterization of the $\omega$-limit sets of interval maps, Acta Math. Hungar., 88 (2000), 291-300. doi: 10.1023/A:1026775906693.
[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), 21-31. doi: 10.1017/S0143385708001089.
[3]	A. D. Barwell, A characterization of $\omega$-limit sets of piecewise monotone maps of the interval, Fund. Math., 207 (2010), 161-174. doi: 10.4064/fm207-2-4.
[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), 1819-1833. doi: 10.3934/dcds.2013.33.1819.
[5]	W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, American Mathematical Society, Providence, R. I., 1955.
[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), 107-131. doi: 10.1023/A:1009044515567.
[7]	M. Hochman, On the dynamics and recursive properties of multidimensional symbolic systems, Invent. Math., 176 (2009), 131-167. doi: 10.1007/s00222-008-0161-7.
[8]	M. Hochman and T. Meyerovitch, A characterization of the entropies of multidimensional shifts of finite type, Ann. of Math. (2), 171 (2010), 2011-2038. doi: 10.4007/annals.2010.171.2011.
[9]	A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995.
[10]	D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302.
[11]	P. Oprocha, Chain recurrence in multidimensional time discrete dynamical systems, Discrete Contin. Dyn. Syst., 20 (2008), 1039-1056. doi: 10.3934/dcds.2008.20.1039.
[12]	P. Oprocha, Shadowing in multi-dimensional shift spaces, Colloq. Math., 110 (2008), 451-460. doi: 10.4064/cm110-2-8.
[13]	P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.

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