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

TL;DR

This paper investigates conditions under which functions in the Ellis semigroup of a discrete dynamical system are continuous, especially in countable metrizable spaces, and explores the structure of the Ellis semigroup in transitive systems.

Contribution

It provides necessary and sufficient conditions for the continuity of Ellis semigroup functions and characterizes the Ellis semigroup in transitive systems where all functions are continuous or accumulation points have singleton omega-limit sets.

Findings

All functions of the Ellis semigroup are continuous if every accumulation point is fixed.

In transitive systems, if all Ellis functions are continuous or accumulation points have singleton omega-limit sets, then the Ellis semigroup is homeomorphic to the phase space.

Examples illustrate the theoretical results.

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 $a\in X'$ 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.