Relativized weak disjointness and relatively invariant measures
Relativized weak disjointness and relatively invariant measures
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DOI：
10.1090/s0002-9947-1978-0467704-9
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发表时间：
1978-02
影响因子：
1.3
通讯作者：
D. Mcmahon
中科院分区：
文献类型：
--
作者：
D. Mcmahon
引言。在[P]中，R. 佩莱格证明了如下内容：如果(X, T)和(Y, T)是支持不变测度的度量极小变换群，那么(X, T)和(Y, T)是弱不交的（(X×Y, T)有一个轨道稠密的点）当且仅当它们的极大等度连续因子(X/S, T)和(Y/S, T)是不交的。受此结果的启发，格拉斯纳在[G]结尾处猜想：如果(X, T)→(Y, T)支持一个相对不变测度，其中X是极小的且是度量空间，那么R(φ) = {(x, x')∈X×X | φ(x) = φ(x')}有一个轨道稠密的点当且仅当Y的唯一作为X的因子的几乎周期扩张就是Y本身。上述“仅当”部分很容易看出是正确的。本文的例3.2.1表明，如果不对φ施加一些进一步的限制，“如果”部分是不正确的。对φ的一个可能的限制是假设它是开的。有了这个附加条件，我们证明了格拉斯纳猜想的如下变体（见1.9）。假设X和Y是极小的且是度量空间，自然投影X→X/S(φ)是开的（其中S(φ)是相对化的等度连续结构关系），φ有一个相对不变测度，并且X/S(φ)和Y/S(φ)是相对不交的（X/S(φ)∘Z Y/S(φ) = {(x/S(φ), y/S(φ)) : φ(x) = φ(y)}是一个极小集）。在X中固定x。那么对于Y中包含的某个非空紧集B(x)，集合D(x) = {y∈B(x) : (x, y)在X∘I Y中有一个稠密轨道}是B(x)的一个稠密的G_δ子集。当Z是一个单点集时，可以去掉X→X/S(φ)是开的这个限制，这样就给我们一个佩莱格定理的推广。我们还表明，如果X是极小的且φ : X→Z有一个相对不变测度并且如果z∈Z，那么存在一个紧集B_z⊂X使得对于x
Introduction. In [P] R. Peleg proves the following: If (X, T) and (Y, T) are metric minimal transformation groups supporting invariant measures then (X, T) and (Y, T) are weakly disjoint ((X x Y, T) has a point with dense orbit) iff their maximal equicontinuous factors (X/S, T) and (Y/S, T) are disjoint. Motivated by this result Glasner conjectures at the end of [G] that If (X, T) -*' (Y, T) supports a relatively invariant measure where X is mnimal and metric, then R((p) = {(x, x') E X x X1(p(x) = p(x')} has a point with dense orbit iff the only almost periodic extension of Y which is a factor of X is Y itself. The only if part of the above is easily seen to be true. Example 3.2.1 of this paper shows that the if part is not true without some further restriction on (p. One possible restriction on p is to assume that it is open. With this additional condition we prove the following variation of Glasner's conjecture (see 1.9). Suppose X and Y are minimal and metric, the natural projection X -* X/SQ() is open (where S(() is the relativized equicontinuous structure relation), 0 has a relative invariant measure, and X/S (p) and Y/S(0) are relatively disjoint (X/S(p) o Z Y/S(0) = {(x/S(q)), y/S(0)): p(x) = 0(y)) is a minimal set). Fix x in X. Then for some nonempty compact set, B (x), contained in Y, the set D (x) = {y e B (x): (x, y) has a dense orbit in X o I Y) is a dense G,8 subset of B(X). When Z is a singleton, the restriction that X-> XIS (p) is open can be dropped, thus giving us a generalization of Peleg's theorem. We also show that if X is minimal and cp: X --* Z has a relatively invariant measure and if z E Z, then there exists a compact set B2 C X such that for x