An uncountable family of pairwise non-Kakutani equivalent smooth diffeomorphisms

We construct an uncountable family of smooth ergodic zero-entropy diffeomorphisms that are pairwise non-Kakutani equivalent, 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. To that end, we first construct a smooth ergodic zero-entropy and non-Loosely Bernoulli diffeomorphism, by suitably modifying a smooth construction by Anosov and Katok. A construction of this kind was announced by Katok in 1977 and 1980 [8, p.141], [9, p.293].