Abstract
Ergodic homeomorphisms T and S of Polish probability spaces X and Y are evenly Kakutani equivalent if there is an orbit equivalence ϕ: X 0 → Y 0 between full measure subsets of X and Y such that, for some A ⊂ X 0 of positive measure, ϕ restricts to a measurable isomorphism of the induced systems T A and S ϕ(A). The study of even Kakutani equivalence dates back to the seventies, and it is well known that any two irrational rotations of the circle are evenly Kakutani equivalent. But even Kakutani equivalence is a purely measurable relation, while systems such as irrational rotations are both measurable and topological.
Recently del Junco, Rudolph and Weiss [1] studied a new relation called nearly continuous Kakutani equivalence. A nearly continuous Kakutani equivalence is an even Kakutani equivalence where also X 0 and Y 0 are invariant G δ sets, A is within measure zero of both open and closed, and ϕ is a homeomorphism from X 0 to Y 0. It is known that nearly continuous Kakutani equivalence is strictly stronger than even Kakutani equivalence and is the natural strengthening of even Kakutani equivalence to the nearly continuous category—the category where maps are continuous after sets of measure zero are removed. In this paper we show that any two irrational rotations of the circle are nearly continuously Kakutani equivalent.
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Partially supported by NSF grant DMS0700874.
Professor Rudolph passed away on February 4, 2010.
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Dykstra, A., Rudolph, D.J. Any two irrational rotations are nearly continuously Kakutani equivalent. JAMA 110, 339–384 (2010). https://doi.org/10.1007/s11854-010-0009-0
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DOI: https://doi.org/10.1007/s11854-010-0009-0