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The structure of pointwise recurrent expansive homeomorphisms

Part of: Topological dynamics

Published online by Cambridge University Press:  03 November 2022

ENHUI SHI ,
HUI XU Open the ORCID record for HUI XU [Opens in a new window]  and
ZIQI YU
ENHUI SHI
Affiliation:
School of Mathematical Sciences, Soochow University, Suzhou, Jiangsu 215006, China (e-mail: ehshi@suda.edu.cn; 20204207013@stu.suda.edu.cn)
HUI XU*
Affiliation:
CAS Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
ZIQI YU
Affiliation:
School of Mathematical Sciences, Soochow University, Suzhou, Jiangsu 215006, China (e-mail: ehshi@suda.edu.cn; 20204207013@stu.suda.edu.cn)
*
e-mail: huixu2734@ustc.edu.cn
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Abstract

Let X be a compact metric space and let $f: X\!\rightarrow \! X$ be a homeomorphism on X. We show that if f is both pointwise recurrent and expansive, then the dynamical system $(X, f)$ is topologically conjugate to a subshift of some symbolic system. Moreover, if f is pointwise positively recurrent, then the subshift is semisimple; a counterexample is given to show the necessity of positive recurrence to ensure the semisimplicity.

Keywords

recurrenceexpansivityalmost periodic pointsymbolic systemtopological dimension

MSC classification

Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 37B20: Notions of recurrence
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 10 , October 2023 , pp. 3448 - 3459
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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