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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 $s,t\in S$, $T_s\circT_t=T_{st}$. Given a point $p\in\beta S$, the Stone-\v Cech compactification ofthe 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 theoperation extending the operation of $S$ such that $\beta S$ is a righttopological semigroup and multiplication on the left by any point of $S$ iscontinuous. Given $p,q\in\beta S$, $T_p\circ T_q=T_{pq}$, but $T_p$ is usuallynot continuous. Given a dynamical system $(X,\langle T_s\rangle_{s\in S})$, anda point $x\in X$, we let $U(x)=\{p\in\beta S:T_p(x)$ is uniformlyrecurrent$\}$. We show that each $U(x)$ is a left ideal of $\beta S$ and forany semigroup we can get a dynamical system with respect to which $K(\betaS)=\bigcap_{x\in X}U(x)$ 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 thateach such $U(x)$ properly contains $K(\beta S)$ and has $U(x) \setminus c\ellK(\beta S)\neq \emptyset$.

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