Continued exploration of nearly continuous Kakutani equivalence

Abstract

Nearly continuous dynamical systems, a relatively new field of study, blends together topological dynamics and measurable dynamics/ergodic theory by asking that properties hold modulo sets both meager and of measure zero. In the measure theoretic category, two dynamical systems (X, T) and (Y, S) are called Kakutani equivalent if there exists measurable subsets A subset of X and B subset of Y such that the induced transformations TA and SB are measurably conjugate. We say that a set A subset of X is nearly clopen if it is clopen in the relative topologyof a dense Gδ subset of full measure. Nearly continuous Kakutani equivalence refines the measure-theoretic notion by requiring the sets A and B to be nearly clopen and TA and SB to be nearly continuously conjugate. If A and B have the same measure, then we say that the systems are nearly continuously evenly Kakutani equivalent. All irrational rotations of the circle and all odometers belong to the same equivalence class for nearly continuous even Kakutani equivalence. For our first main result, we prove that if A and B are nearly clopen subsets of the same measure of X and Y respectively, and if ϕ is a nearly continuous conjugacy between TA and SB, then ϕ extends to a nearly continuous orbit equivalence between T and S. We also prove that if A subset of X and B subset of Y are nearly clopen sets such that the measure of A is larger than the measure of B, and if T is a nearly uniquely ergodic transformation and TA is nearly continuously conjugate to SB, then there exists B' subset of Y such that X is nearly continuously conjugate to SB'. We then introduce the natural topological analog of rank one transformations, called strongly rank one transformations, and show that all strongly rank one transformations are nearly continuously evenly Kakutani equivalent to the class containing all adding machines. Our main result proves that all minimal isometries of compact metric spaces are nearly continuously evenly Kakutani equivalent to the binary odometer.