Recurrence in the dynamical system (X,〈Ts〉s∈S) and ideals of βS

A dynamical system is a pair (X,〈Ts〉s∈S), where X is a compact Hausdorff space, S is a semigroup, for each s∈S, Ts is a continuous function from X to X, and for all s,t∈S, Ts∘Tt=Tst. Given a point p∈βS, the Stone-Čech compactification of the discrete space S, Tp:X→X is defined by, for x∈X, Tp(x)=p−lims∈STs(x). We let βS have the operation extending the operation of S such that βS is a right topological semigroup and multiplication on the left by any point of S is continuous. Given p,q∈βS, Tp∘Tq=Tpq, but Tp is usually not continuous. Given a dynamical system (X,〈Ts〉s∈S), and a point x∈X, we let U(x)={p∈βS:Tp(x) be uniformly recurrent}. We show that each U(x) is a left ideal of βS and for any semigroup we can get a dynamical system with respect to which K(βS)=⋂x∈XU(x) and cℓK(βS)=⋂{U(x):x∈XandU(x)is closed}. And we show that weak cancellation assumptions guarantee that each such U(x) properly contains K(βS) and has U(x)∖cℓK(βS)≠∅.