Symbolic topological dynamics in the circle

Morales, C.; Oh, Jumi

doi:10.1090/proc/14472

Symbolic topological dynamics in the circle
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by C. A. Morales and Jumi Oh PDF

Proc. Amer. Math. Soc. 147 (2019), 3413-3424 Request permission

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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