Open AccessSymbolic topological dynamics in the circle
Author(s) -
C. A. Morales,
Jumi Oh
Publication year - 2019
Publication title -
proceedings of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.968
H-Index - 84
eISSN - 1088-6826
pISSN - 0002-9939
DOI - 10.1090/proc/14472
Subject(s) - computer science , symbolic dynamics , expansive , homeomorphism (graph theory) , topological entropy , algorithm , mathematics , discrete mathematics , pure mathematics , materials science , compressive strength , composite material
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 N -expansivity, and symbolic shadowing property . We analyze dynamical systems with these properties in the circle. Indeed, we show that every symbolically N 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.