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Expansive dynamics on locally compact groups

Part of: Topological dynamics Connections with other structures, applications

Published online by Cambridge University Press:  18 November 2020

BRUCE P. KITCHENS
BRUCE P. KITCHENS*
Affiliation:
Indiana University–Purdue University Indianapolis, Indianapolis, IN46202, USA
*
e-mail: bkitchen@iupui.edu
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Abstract

Let $\mathcal {G}$ be a second countable, Hausdorff topological group. If $\mathcal {G}$ is locally compact, totally disconnected and T is an expansive automorphism then it is shown that the dynamical system $(\mathcal {G}, T)$ is topologically conjugate to the product of a symbolic full-shift on a finite number of symbols, a totally wandering, countable-state Markov shift and a permutation of a countable coset space of $\mathcal {G}$ that fixes the defining subgroup. In particular if the automorphism is transitive then $\mathcal {G}$ is compact and $(\mathcal {G}, T)$ is topologically conjugate to a full-shift on a finite number of symbols.

Keywords

automorphismslocally compact groupssymbolic dynamics

MSC classification

Primary: 37B10: Symbolic dynamics
Secondary: 54H11: Topological groups
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 12 , December 2021 , pp. 3768 - 3779
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Cornulier, Y. and de la Harpe, P.. Metric Geometry of Locally Compact Groups ( European Mathematical Society Tracts in Mathematics , 25). European Mathematical Society, Zurich, 2016.CrossRefGoogle Scholar
Caprace, P.-E. and Monod, N. (Eds.). New Directions in Locally Compact Groups ( London Mathematical Society Lecture Note Series , 447). Cambridge University Press, Cambridge, 2018.CrossRefGoogle Scholar
Handel, M., Kitchens, B. P. and Rudolph, D. J.. Metrics and entropy for non-compact sets. Israel J. Math. 91 (1995), 253–271.CrossRefGoogle Scholar
Kitchens, B. P.. Expansive dynamics on zero-dimensional groups. Ergod. Th. & Dynam. Sys. 7 (1987), 249–261.CrossRefGoogle Scholar
Kitchens, B. P.. Symbolic Dynamics; One sided, Two-sided and Countable State Markov Shifts. Springer, Berlin, 1998.Google Scholar