Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8906112 | Indagationes Mathematicae | 2018 | 20 Pages |
Abstract
A dynamical system is a pair (X,ãTsãsâS), where X is a compact Hausdorff space, S is a semigroup, for each sâS, Ts is a continuous function from X to X, and for all s,tâS, TsâTt=Tst. Given a point pâβS, the Stone-Äech compactification of the discrete space S, Tp:XâX is defined by, for xâX, Tp(x)=pâlimsâSTs(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,qâβS, TpâTq=Tpq, but Tp is usually not continuous. Given a dynamical system (X,ãTsãsâS), and a point xâX, we let U(x)={pâβS: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)=âxâXU(x) and câK(βS)=â{U(x):xâXandU(x)is closed}. And we show that weak cancellation assumptions guarantee that each such U(x) properly contains K(βS) and has U(x)âcâK(βS)â â
.
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Neil Hindman, Dona Strauss, Luca Q. Zamboni,