Analytic pseudo-rotations

TL;DR

This paper constructs specific analytic symplectomorphisms on the cylinder and sphere with few periodic points, demonstrating new dynamical behaviors and disproving longstanding conjectures, while introducing a novel approximation theorem.

Contribution

It develops a new approximation theorem enabling the implementation of the Anosov-Katok scheme in analytic settings and constructs symplectomorphisms with unique properties.

Findings

Constructed symplectomorphisms with zero or two periodic points

Disproved Birkhoff's conjecture from 1941

Solved a problem posed by Herman in 1998

Abstract

We construct analytic symplectomorphisms of the cylinder or the sphere with zero or exactly two periodic points and which are not conjugated to a rotation. In the case of the cylinder, we show that these symplectomorphisms can be chosen ergodic or to the contrary with local emergence of maximal order. In particular, this disproves a conjecture of Birkhoff (1941) and solve a problem of Herman (1998). One aspect of the proof provides a new approximation theorem, it enables in particular to implement the Anosov-Katok scheme in new analytic settings.