Pointed minimal sets and 𝑆-regularity

Abstract

Let (X, T) be a point transitive transformation group (Cl (xT) = X for some xE X) with compact Hausdorff phase space X and discrete phase group T. Let 3T be the Stone-Cech compactification of T. If tET and {Sm} is a net in T converging to pGEEOT, put pt=lim Smt. If further { t, } is a net in T converging to q fEO3T, put pq =lim ptn. Ellis has observed [3] that these definitions make (fT, T) into a transformation group and make 3T into a semigroup in which left multiplication is continuous but right multiplication is generally not. Let e(/3T) = e be the algebra of all real-valued continuous functions on fT. Forf E e and t E T letfRt be the element of e, given by (fRt, p) = (f, pt) (pEE3T) (generally (f, p) =pf=the image of p under f). Define a subalgebra a of e to be a T-subalgebra if it is uniformly closed and iffRtEl whenever fEa and tET. Call a homomorphism of a into e a T-homomorphism if it commutes with Rt for all tET, and denote the set of all T-homomorphisms on a by a a . For eachfaE and tE T, define the map af t of I a I into the reals by O = (fc, t) (p E Ie a I). Ellis has shown [4] that, if I a I is provided with the smallest topology such that all the maps af t are continuous, then the action defined by (f(pt), p)= (fc, tp) (f E a, ck: E |I, tET, p C-3T) makes (I aI, T) into a point transitive transformation group for each T-subalgebra a. Further, given (X, T), a T-subalgebra a can be found such that ( a| , T) is isomorphic to (X, T) (we say a corresponds to X). In general there are many T-subalgebras corresponding to a given X. For convenience we repeat the Ellis construction, without proof. Choose xEX such that Cl(xT) =X. The map 7r. of Tonto X given by t7r =xt (t E T) becomes, when extended continuously, a homomorphism 7r * of (fT, T) onto (X, T); that is, a continuous map commuting with T. Write xp for p7r * (pEi3T), and define a map x* from e(X) into e by (fx*, p)=(f, xp) (fEC(X), pEfiT). Then a. = e(X)x* corresponds to X. Further, there is a natural isomorphisrr between (I axI, T) and (X, T) taking the inclusion map of I ax| onto x. Following Ellis, we shall say that a T-subalgebra (a of e has a certain recursive property if (I al , T) has that property. We shall primarily consider minimal algebras. For f e, q EofT, define fq ( e by (fq, p) = (f, qp) (p COT). Then Ellis has shown [5] that a is mini-