On pseudo-rotations of the annulus with generic rotation number

TL;DR

This paper proves that for a generic rotation number, area-preserving smooth pseudo-rotations of the annulus are limits of rotations, and generically exhibit weak mixing behavior.

Contribution

It establishes that for a Baire generic rotation number, pseudo-rotations are approximated by conjugates of rotations, revealing generic dynamical properties.

Findings

Pseudo-rotations with generic rotation number are limits of conjugates of rotations.

A generic pseudo-rotation with a generic rotation number is weakly mixing.

The set of such pseudo-rotations equals the closure of conjugates of rotations.

Abstract

We show that for a Baire generic rotation number $\alpha \in \mathbb{R} / \mathbb{Z}$, the set of area preserving $C^\infty$-pseudo-rotations of the annulus $\mathbb{A}$ with rotation number $\alpha$ equals the closure of the set of area preserving $C^\infty$-pseudo-rotations which are smoothly conjugate to the rotation $R_{\alpha}$. As a corollary, a $C^\infty$-generic area preserving pseudo-rotation of the annulus with a Baire generic rotation number $\alpha$ is weakly mixing.