MEASURED TOPOLOGICAL ORBIT AND KAKUTANIEQUIVALENCE

ANDRES DEL JUNCO; DANIEL J. RUDOLPH; BENJAMIN WEISS

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MEASURED TOPOLOGICAL ORBIT AND KAKUTANIEQUIVALENCE

机译：MEASURED TOPOLOGICAL ORBIT AND KAKUTANIEQUIVALENCE

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Suppose X and Y are Polish spaces each endowed with Borel prob-ability measures and We call these Polish probability spaces. We say amap 4) is a nearly continuous if there are measurable subsets X0 C X andY0 C Y, each of full measure, and : X0 –4 Y0 is measure-preserving andcontinuous in the relative topologies on these subsets. We show that this isa natural context to study morphisms between ergodic homeomorphisms ofPolish probability spaces. In previous work such maps have been called almostcontinuous or finitary. We propose the name measured topological dynamicsfor this area of study. Suppose one has measure-preserving and ergodic mapsT and S acting on X and Y respectively. Suppose cb is a measure-preservingbijection defined between subsets of full measure on these two spaces. Ourmain result is that such a (15. can always be regularized in the following sense.Both T and S have full groups (FG(T) and FG(S)) consisting of those mea-surable bijections that carry a point to a point on the same orbit. We willshow that there exists f E FG(T) and h E FG(S) so that NV is nearly con-tinuous. This comes close to giving an alternate proof of the result of delJunco and *ahin, that any two measure-preserving ergodic homeomorphismsof nonatomic Polish probability spaces are continuously orbit equivalent oninvariant G,5 subsets of full measure. One says T and S are evenly Kakutaniequivalent if one has an orbit equivalence cb which restricted to some subsetis a conjugacy of the induced maps. Our main result implies that any suchmeasurable Kakutani equivalence can be regularized to a Kakutani equivalencethat is nearly continuous. We describe a natural nearly continuous analogue ofKakutani equivalence and prove it strictly stronger than Kakutani equivalence.To do this we introduce a concept of nearly unique ergodicity.

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来源
《Discrete and Continuous Dynamical Systems, Series S》 |2009年第2期|221-238|共18页
作者
ANDRES DEL JUNCO; DANIEL J. RUDOLPH; BENJAMIN WEISS;
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作者单位

Department of Mathematics, University of TorontoToronto, Ontario, Canada M5S 2E4;

Department of Mathematics, Colorado State UniversityFort Collins, CO 80523-1874, USA;

Einstein Institute of MathematicsEdmond J. Safra Campus, Givat RamThe Hebrew University of JerusalemJerusalem, 91904, Israel;

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正文语种英语
中图分类数学;
关键词
Measured Topological Dynamics; orbit equivalence; Kakutani equiva-lence; finitary.;

MEASURED TOPOLOGICAL ORBIT AND KAKUTANIEQUIVALENCE

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