Iterates of dynamical systems on compact metrizable countable spaces

TL;DR

This paper investigates the structure of Ellis semigroups in dynamical systems on compact metrizable countable spaces, revealing conditions under which functions are continuous or discontinuous, and providing a specific example illustrating mixed behavior.

Contribution

It characterizes the continuity properties of Ellis semigroup functions in countable compact metric spaces and constructs an example with mixed continuity.

Findings

If all accumulation points are periodic, then all functions in $E(X,f)^*$ are either continuous or discontinuous.

An example exists where $E(X,f)^*$ contains both continuous and discontinuous functions.

The structure of $E(X,f)$ depends on the periodicity of accumulation points.

Abstract

Given a dynamical system $(X,f)$, we let $E(X,f)$ denote its Ellis semigroup and $E(X,f)^* = E(X,f) \setminus \{f^n : n \in \mathbb{N}\}$. We analyze the Ellis semigroup of a dynamical system having a compact metric countable space as a phase space. We show that if $(X,f)$ is a dynamical system such that $X$ is a compact metric countable space and every accumulation point $X'$ is periodic, then either each function of $E(X,f)^*$ is continuous or each function of $E(X,f)^*$ is discontinuous. We describe an example of a dynamical system $(X,f)$ where $X$ is a compact metric countable space, the orbit of each accumulation point is finite and $E(X,f)^*$ contains continuous and discontinuous functions.