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Proximal relations in topological dynamics

1965; American Mathematical Society; Volume: 16; Issue: 3 Linguagem: Inglês

10.1090/s0002-9939-1965-0179775-4

1088-6826

T. S. Wu,

Topological and Geometric Data Analysis

In this note we shall prove that when the proximal relation of a transformation group (X, T, it) with compact phase space X is transitive (i.e., it is an equivalence relation), then it is equivalent with the syndetically proximal relation.This would answer two questions in [4, Remark 5].Standing notations.Let (X, T, ir) be a transformation group with compact phase space.The proximal relation of (X, T, if) is denoted by P(X) and the syndetically proximal relation by L(X).The product transformation group induced by (X, P, 7r):;will be denoted by (XXX, T, p), which is defined by (x, y)pi = (xirt, yir1) for (x, y) QXXX and tQ T. For simplicity we shall write xt for X7r' and (xt, yt) = (x, y)t for (x, y)p'.Reference.The proximal relation was studied in [l], [2], [3], [4].The syndetically proximal relation was defined and studied in [4].Proposition.If P(X) is transitive, then P(X)=L(X).Proof.Since P(X) is transitive, so is P(XXX)[l].Then each orbit closure Cl(x, y)T in (XXX, P, p) contains a unique minimal set.Let (x, y)QP(X).If Cl(x, y)T-P(X)^n, then there is (a, b) £Cl(x, y)T-P(X).Let M be the (unique) minimal set contained in Cl(a, b) T.There are two cases.Case 1. MC\P(X) = □■ By Lemma 2 of [ l], there is a point (u,v)QM such that ((x, y), (u, v))QP(XXX).This shows that (x, u)QP(X), (y, v)QP(X), a fortiori, (u, v)QP(X) by the transitivity of P(X).We have the contradiction.Case 2. MC\P(X) * □• By the definition of P(X), if (x', /) £P(X) and N is the minimal set contained in Cl(x', y')T, then NQA(X), the diagonal of XXX.This shows that MQA(X), a fortiori, (a, b) £P(X).We have the contradiction also.Hence, Cl(x, y)PCP(X) when (x, y)QP(X).By Lemma 5 of [4], P(X) C7(X), P(X) =P(X).Corollary.P(X) is an equivalence relation if and only if P(XXX) =P(X)XP(X).