Smooth Pseudo-Rotations Measure-Theoretically Isomorphic to Circle Rotations of Rationally Independent Angle

TL;DR

The paper constructs smooth volume-preserving diffeomorphisms on manifolds that are metrically isomorphic to irrational circle rotations, expanding understanding of pseudo-rotations and their measure-theoretic properties.

Contribution

It demonstrates the existence of smooth pseudo-rotations on manifolds that are measure-theoretically isomorphic to irrational circle rotations, with specific boundary behavior.

Findings

Existence of smooth ergodic pseudo-rotations on the annulus

Pseudo-rotations are metrically isomorphic to irrational circle rotations

Boundary tangent property of the constructed pseudo-rotations

Abstract

Let M be a smooth compact connected manifold, on which there exists an effective smooth circle action preserving a positive smooth volume. We show that on M, the smooth closure of the smooth volume-preserving conjugation class of some Liouville rotations of angle alpha contains a smooth volume-preserving diffeomorphism T that is metrically isomorphic to an irrational rotation of angle beta on the circle, with alpha different of beta, and with alpha and beta chosen either rationally dependent or rationally independent. In particular, if M is the closed annulus, M admits a smooth ergodic pseudo-rotation T of angle alpha that is metrically isomorphic to the rotation of angle beta. Moreover, T is smoothly tangent to the rotation of angle alpha on the boundary of M.