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Proceedings of the American Mathematical Society

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Symbolic topological dynamics in the circle

Authors: C. A. Morales and Jumi Oh
Journal: Proc. Amer. Math. Soc. 147 (2019), 3413-3424
MSC (2010): Primary 37B05; Secondary 54H20
Published electronically: March 26, 2019
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Abstract: We explain how dynamical systems with generating partitions are symbolically expansive, namely symbolic counterparts of the expansive ones. Similar ideas allow the notions of symbolic equicontinuity, symbolic distality, symbolic $ N$-expansivity, and symbolic shadowing property. We analyze dynamical systems with these properties in the circle. Indeed, we show that every symbolically $ N$-expansive circle homeomorphism has finitely many periodic points. Moreover, if there are no wandering points, then the situation will depend on the rotation number. In the rational case the homeomorphism is symbolically equicontinuous with the symbolic shadowing property and, in the irrational case, the homeomorphism is symbolically expansive, symbolically distal, but not symbolically equicontinuous. We will also introduce a symbolic entropy and study its properties.

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Additional Information

C. A. Morales
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530 21945-970, Rio de Janeiro, Brazil

Jumi Oh
Affiliation: Department of Mathematics, Sungkyunkwan University, Suwon, 16419, Republic of Korea

Keywords: Symbolically expansive, symbolic shadowing, metric space
Received by editor(s): July 3, 2018
Received by editor(s) in revised form: November 3, 2018, and November 4, 2018
Published electronically: March 26, 2019
Additional Notes: The first author was partially supported by CNPq-Brazil-303389/2015-0 and the NRF Brain Pool Grant funded by the Korea government (No. 2018H1D3A2001632).
The second author was supported by the National Research Foundation of Korea (NRF) grant funded by the MEST 2015R1A3A2031159 and 2016R1D1A1B03931962.
Communicated by: Wenxian Shen
Article copyright: © Copyright 2019 American Mathematical Society