On manifolds with smooth circle actions, there exist smooth volume-preserving diffeomorphisms metrically isomorphic to circle rotations of a different, possibly rationally independent angle.
An uncountable family of pairwise no n-Kakutani equivalent smooth di ffeomorphisms
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Smooth Pseudo-Rotations Measure-Theoretically Isomorphic to Circle Rotations of Rationally Independent Angle
On manifolds with smooth circle actions, there exist smooth volume-preserving diffeomorphisms metrically isomorphic to circle rotations of a different, possibly rationally independent angle.