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

Neil Hindman, Dona Strauss, Luca Q. Zamboni

Published 2016-08-19Version 1

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 $U(x) \setminus c\ell K(\beta S)\neq \emptyset$.