On the continuity of the elements of the Ellis semigroup and other properties

García-Ferreira, Salvador, Rodríguez-López, Yackelin, and Uzcátegui, Carlos. "On the continuity of the elements of the Ellis semigroup and other properties." Commentationes Mathematicae Universitatis Carolinae (2021): 225-241. <http://eudml.org/doc/297476>.

@article{García2021,
abstract = {We consider discrete dynamical systems whose phase spaces are compact metrizable countable spaces. In the first part of the article, we study some properties that guarantee the continuity of all functions of the corresponding Ellis semigroup. For instance, if every accumulation point of $X$ is fixed, we give a necessary and sufficient condition on a point $a\in X^\{\prime \}$ in order that all functions of the Ellis semigroup $E(X,f)$ be continuous at the given point $a$. In the second part, we consider transitive dynamical systems. We show that if $(X,f)$ is a transitive dynamical system and either every function of $E(X,f)$ is continuous or $|\omega _f(x)|=1$ for each accumulation point $x$ of $X$, then $E(X,f)$ is homeomorphic to $X$. Several examples are given to illustrate our results.},
author = {García-Ferreira, Salvador, Rodríguez-López, Yackelin, Uzcátegui, Carlos},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {discrete dynamical system; Ellis semigroup; $p$-iterate; $p$-limit point; ultrafilter; compact metric countable space},
language = {eng},
number = {2},
pages = {225-241},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the continuity of the elements of the Ellis semigroup and other properties},
url = {http://eudml.org/doc/297476},
year = {2021},
}

TY - JOUR
AU - García-Ferreira, Salvador
AU - Rodríguez-López, Yackelin
AU - Uzcátegui, Carlos
TI - On the continuity of the elements of the Ellis semigroup and other properties
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2021
PB - Charles University in Prague, Faculty of Mathematics and Physics
IS - 2
SP - 225
EP - 241
AB - We consider discrete dynamical systems whose phase spaces are compact metrizable countable spaces. In the first part of the article, we study some properties that guarantee the continuity of all functions of the corresponding Ellis semigroup. For instance, if every accumulation point of $X$ is fixed, we give a necessary and sufficient condition on a point $a\in X^{\prime }$ in order that all functions of the Ellis semigroup $E(X,f)$ be continuous at the given point $a$. In the second part, we consider transitive dynamical systems. We show that if $(X,f)$ is a transitive dynamical system and either every function of $E(X,f)$ is continuous or $|\omega _f(x)|=1$ for each accumulation point $x$ of $X$, then $E(X,f)$ is homeomorphic to $X$. Several examples are given to illustrate our results.
LA - eng
KW - discrete dynamical system; Ellis semigroup; $p$-iterate; $p$-limit point; ultrafilter; compact metric countable space
UR - http://eudml.org/doc/297476
ER -