Measured topological orbit and Kakutani equivalence

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An example of Kakutani equivalent and strong orbit equivalent substitution systems that are not conjugate

June 2009, 2(2): 221-238. doi: 10.3934/dcdss.2009.2.221

Measured topological orbit and Kakutani equivalence

Andres del Junco ^1, , Daniel J. Rudolph ^2, and Benjamin Weiss ^3,

1.	Department of Mathematics, University of Toronto, Toronto, Ontario, Canada
2.	Department of Mathematics, Colorado State University, Fort Collins, CO 80523, United States
3.	Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904

Received February 2008 Revised October 2008 Published April 2009

Abstract
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Suppose $X$ and $Y$ are Polish spaces each endowed with Borel probability measures $\mu$ and $\nu$. We call these Polish probability spaces. We say a map $\phi$ is a nearly continuous if there are measurable subsets $X_0\subseteq X$ and $Y_0\subseteq Y$, each of full measure, and $\phi:X_0\to Y_0$ is measure-preserving and continuous in the relative topologies on these subsets. We show that this is a natural context to study morphisms between ergodic homeomorphisms of Polish probability spaces. In previous work such maps have been called almost continuous or finitary. We propose the name measured topological dynamics for this area of study. Suppose one has measure-preserving and ergodic maps $T$ and $S$ acting on $X$ and $Y$ respectively. Suppose $\phi$ is a measure-preserving bijection defined between subsets of full measure on these two spaces. Our main result is that such a $\phi$ can always be regularized in the following sense. Both $T$ and $S$ have full groups ($FG(T)$ and $FG(S)$) consisting of those measurable bijections that carry a point to a point on the same orbit. We will show that there exists $f\in FG(T)$ and $h\in FG(S)$ so that $h\phi f$ is nearly continuous. This comes close to giving an alternate proof of the result of del Junco and Şahin, that any two measure-preserving ergodic homeomorphisms of nonatomic Polish probability spaces are continuously orbit equivalent on invariant $G_\delta$ subsets of full measure. One says $T$ and $S$ are evenly Kakutani equivalent if one has an orbit equivalence $\phi$ which restricted to some subset is a conjugacy of the induced maps. Our main result implies that any such measurable Kakutani equivalence can be regularized to a Kakutani equivalence that is nearly continuous. We describe a natural nearly continuous analogue of Kakutani equivalence and prove it strictly stronger than Kakutani equivalence. To do this we introduce a concept of nearly unique ergodicity.

Keywords: Kakutani equivalence, finitary, orbit equivalence, Measured Topological Dynamics.

Mathematics Subject Classification: Primary: 37A05, 37A20 ; Secondary: 37A15, 54H2.

Citation: Andres del Junco, Daniel J. Rudolph, Benjamin Weiss. Measured topological orbit and Kakutani equivalence. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 221-238. doi: 10.3934/dcdss.2009.2.221

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