The Anosov-Katok method and pseudo-rotations in symplectic dynamics

TL;DR

This paper demonstrates the existence of Hamiltonian pseudo-rotations with minimal ergodic measures on toric symplectic manifolds using the Anosov-Katok conjugation method, revealing specific measure structures.

Contribution

It introduces a novel application of the Anosov-Katok method to construct pseudo-rotations with minimal ergodic measures in symplectic geometry.

Findings

Existence of pseudo-rotations with exactly two ergodic measures.

The ergodic measures are the symplectic volume measure and fixed point measures.

Construction relies on conjugation method of Anosov and Katok.

Abstract

We prove that toric symplectic manifolds admit Hamiltonian pseudo-rotations with a finite, and in a sense minimal, number of ergodic measures. The set of ergodic measures of these pseudo-rotations consists of the measure induced by the symplectic volume form and the Dirac measures supported at the fixed points of the torus action. Our construction relies on the conjugation method of Anosov and Katok.