On pseudo-rotations of the annulus with generic rotation number

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.

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