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In ergodic theory, two systems are Kakutani equivalent if there exists aconjugacy between induced transformations. In Measured Topological Orbit andKakutani Equivalence, del Junco, Rudolph, and Weiss defined nearly continuouseven Kakutani equivalence as an orbit equivalence which restricts to aconjugacy between induced maps on nearly clopen sets. This paper defines nearlycontinuous even Kakutani equivalence between two nearly continuous dynamicalsystems as a nearly continuous conjugacy between induced maps on nearly clopensets of the same size and shows that this definition is equivalent to thedefinition given by del Junco, Rudolph, and Weiss. The paper shows that, withan added restriction, if two systems are nearly continuously (non-even)Kakutani equivalent, then one system is isomorphic to an induced transformationof the other, and uses this result to prove that a class of transformationscalled strongly rank one and containing Chacon's map belongs to the same nearlycontinuous even Kakutani equivalence class as irrational rotations.

Nearly Continuous Even Kakutani Equivalence of Strongly Rank One Transformations