Analytic symplectomorphisms displaying minimal ergodicity on the sphere, cylinder and disk

TL;DR

This paper constructs explicit analytic symplectomorphisms on common surfaces that exhibit minimal ergodicity with only three ergodic measures, expanding the understanding of ergodic behavior in symplectic dynamics.

Contribution

It introduces a generalized approach based on the Approximation by Conjugacy method to produce minimally ergodic symplectomorphisms on the sphere, disk, and cylinder.

Findings

Constructed explicit examples of minimally ergodic symplectomorphisms

Extended the Anosov-Katok approximation technique to new settings

Demonstrated minimal ergodicity with only three ergodic measures

Abstract

We construct analytic symplectomorphisms on the sphere, the disk and the cylinder which are minimally ergodic (only 3 ergodic measures). To achieve this, we apply and generalize a principle introduced by Berger, based on the Approximation by Conjugacy method of Anosov-Katok.