Homomorphisms of Transformation Groups

Abstract

Introduction.Let iX, T) he a transformation group with compact phase space X and with arbitrary phase group T. We first point out in Theorem 1 that there exist least invariant closed equivalence relations 5<j and St in X such that T is distal on X\ Sa and T is equicontinuous on X\ Se.This permits, so to speak, the dividing out of certain more complicated parts of a transformation group.The application of this process to properties other than distal and equicontinuous is indicated by Remark 8. Theorem 2 then relates the structure relations 5<j and S, with the proximal and regionally proximal relations of iX, T).Theorem 3 says that these four relations all coincide if (X, T) is locally almost periodic.The concluding remarks show how any transformation group with compact phase space and noncompact phase group gives rise in a natural way to a compact topological group, called its structure group.Such transformation groups, in particular minimal sets, are thus partially classifiable according to their structure groups.As a general reference for the notions occurring here, consult [4].Definition 1.Let (X, /, w) and (F, /, p) be transformation groups with the same phase group T. A homomorphism of (A", T, tt) into or onto (Y, T, p) is defined to be a continuous map d> of X into or onto Y such that tET implies 7r't6=c/>p', or in the condensed notation, such that xEX and tET implies xUp = x<pt.A homomorphism which is at the same time a homeomorphism is called an isomorphism.Of course, any intrinsic property of transformation groups is preserved under isomorphisms onto.See [4, 12.51 and 12.54] for a nontrivial example of a homomorphism taken from symbolic dynamics.Standing notation.Throughout (X, T, tt) and (Y, T, p) will be transformation groups with compact (bicompact Hausdorff) phase spaces X and Y, and <p will be a homomorphism of (X, T) onto (F, /).Definition 2. If/: M-+N, then/: MXM-+NXN is defined by imu m2)f = imxf, m2f) for (mi, m2)EMXM; when no ambiguity is possible, we sometimes write («i, m2)f instead of (wii, mf)f.If FENM, then F denotes [/|/£F];when no ambiguity is possible, we sometimes write AF in place of AF = [af\aEA and/CF] where AEMXM.