Cardinality of the Ellis semigroup on compact metric countable spaces

TL;DR

This paper investigates the size and structure of the Ellis semigroup for dynamical systems on compact countable metric spaces, providing characterizations and examples related to continuity, density, and algebraic properties.

Contribution

It characterizes when the Ellis semigroup and its subset are finite, and explores conditions under which the semigroup has continuum size or is constrained by the space's size.

Findings

If the periodic points have infinitely many periods, the Ellis semigroup has size continuum.

For systems with a dense orbit and continuous Ellis semigroup elements, the size is at most the space's size.

Examples show systems where the Ellis semigroup is homeomorphic but not algebraically homeomorphic to the space.

Abstract

Let $E(X,f)$ be the Ellis semigroup of a dynamical system $(X,f)$ where $X$ is a compact metric space. We analyze the cardinality of $E(X,f)$ for a compact countable metric space $X$. A characterization when $E(X,f)$ and $E(X,f)^* = E(X,f) \setminus \{ f^n : n \in \mathbb{N}\}$ are both finite is given. We show that if the collection of all periods of the periodic points of $(X,f)$ is infinite, then $E(X,f)$ has size $2^{\aleph_0}$. It is also proved that if $(X,f)$ has a point with a dense orbit and all elements of $E(X,f)$ are continuous, then $|E(X,f)| \leq |X|$. For dynamical systems of the form $(\omega^2 +1,f)$, we show that if there is a point with a dense orbit, then all elements of $E(\omega^2+1,f)$ are continuous functions. We present several examples of dynamical systems which have a point with a dense orbit. Such systems provide examples where $E(\omega^2+1,f)$ and $\omega^2+1$ are homeomorphic but not algebraically homeomorphic, where $\omega^2+1$ is taken with the usual ordinal addition as semigroup operation.