Nearly Continuous Even Kakutani Equivalence of Strongly Rank One Transformations

TL;DR

This paper introduces a new notion of nearly continuous even Kakutani equivalence in ergodic theory, proves its equivalence to existing definitions, and shows that strongly rank one transformations, including Chacon's map, are equivalent to irrational rotations under this framework.

Contribution

It defines nearly continuous even Kakutani equivalence, proves its equivalence to previous definitions, and classifies strongly rank one transformations as equivalent to irrational rotations.

Findings

Nearly continuous even Kakutani equivalence is equivalent to the original definition.

Strongly rank one transformations are in the same equivalence class as irrational rotations.

The paper extends the classification of transformations using this new equivalence.

Abstract

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