Recurrence in the dynamical system $(X,\langle T_s\rangle_{s\in S})$ and ideals of $\beta S$

TL;DR

This paper explores the relationship between recurrence in dynamical systems and the algebraic structure of ideals within the Stone-ech compactification of a semigroup, revealing new connections and properties.

Contribution

It establishes that sets of uniformly recurrent points form left ideals in ech compactification semigroups and characterizes their intersections with minimal ideals.

Findings

Each set of uniformly recurrent points forms a left ideal in ech semigroup.

The minimal ideal is the intersection of all such recurrent point sets.

Weak cancellation conditions ensure these sets properly contain the minimal ideal.

Abstract

A {\it dynamical system\/} is a pair $(X,\langle T_s\rangle_{s\in S})$, where $X$ is a compact Hausdorff space, $S$ is a semigroup, for each $s\in S$, $T_s$ is a continuous function from $X$ to $X$, and for all $s,t\in S$, $T_s\circ T_t=T_{st}$. Given a point $p\in\beta S$, the Stone-\v Cech compactification of the discrete space $S$, $T_p:X\to X$ is defined by, for $x\in X$, $\displaystyle T_p(x)=p{-}\!\lim_{s\in S}T_s(x)$. We let $\beta S$ have the operation extending the operation of $S$ such that $\beta S$ is a right topological semigroup and multiplication on the left by any point of $S$ is continuous. Given $p,q\in\beta S$, $T_p\circ T_q=T_{pq}$, but $T_p$ is usually not continuous. Given a dynamical system $(X,\langle T_s\rangle_{s\in S})$, and a point $x\in X$, we let $U(x)=\{p\in\beta S:T_p(x)$ is uniformly recurrent$\}$. We show that each $U(x)$ is a left ideal of $\beta S$ and for any semigroup we can get a dynamical system with respect to which $K(\beta S)=\bigcap_{x\in X}U(x)$ and $c\ell K(\beta S)=\bigcap\{U(x):x\in X$ and $U(x)$ is closed$\}$. And we show that weak cancellation assumptions guarantee that each such $U(x)$ properly contains $K(\beta S)$ and has $U(x) \setminus c\ell K(\beta S)\neq \emptyset$.