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JMD

One says that two ergodic systems $(X,\mathcal F,\mu)$ and $(Y,\mathcal
G,\nu)$ preserving a probability measure are evenly Kakutani equivalent if
there exists an orbit equivalence $\phi: X\to Y$ such that, restricted
to some subset $A\subseteq X$ of positive measure, $\phi$ becomes a
conjugacy between the two induced maps $T_A$ and $S_{\phi(A)}$. It follows
from the general theory of loosely Bernoulli systems developed in [8] that all adding machines are evenly Kakutani equivalent, as they
are rank-1 systems. Recent work has shown that, in systems that are
both topological and measure-preserving, it is natural to seek to
strengthen purely measurable results to be "nearly continuous''. In the
case of even Kakutani equivalence, what one asks is that the map $\phi$ and
its inverse should be continuous on $G_\delta$ subsets of full measure and
that the set $A$ should be within measure zero of being open and of being
closed. What we will show here is that any two adding machines are indeed
equivalent in this nearly continuous sense.

DCDS-S

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.## Year of publication

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