An uncountable family of pairwise non-Kakutani equivalent smooth diffeomorphisms

Abstract

We construct on any smooth compact connected manifold of dimension greater than two on which there exists an effective smooth circle action preserving a positive smooth volume an uncountable family of smooth ergodic zero-entropy diffeomorphisms that are pairwise non-Kakutani equivalent. We first construct a smooth ergodic zero-entropy and non-loosely Bernoulli diffeomorphism by suitably modifying a smooth construction by Anosov and Katok.