Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary

Abstract

Let $M$ be an $m$-dimensional differentiable manifold with a nontrivial circle action {\mathcal S}= {\lbrace S_t \rbrace}_{t \in\RR}, S_{t+1}=S_t, preserving a smooth volume $\mu$. For any Liouville number \a we construct a sequence of area-preserving diffeomorphisms $H_n$ such that the sequence H_n\circ S_\a\circ H_n^{-1} converges to a smooth weak mixing diffeomorphism of $M$. The method is a quantitative version of the approximation by conjugations construction introduced in \cite{AK}. For $m=2$ and $M$ equal to the unit disc \DD^2=\{x^2+y^2\leq 1\} or the closed annulus \AAA=\TT\times [0,1] this result proves the following dichotomy: \a \in \RR \setminus\QQ is Diophantine if and only if there is no ergodic diffeomorphism of $M$ whose rotation number on the boundary equals $\alpha$ (on at least one of the boundaries in the case of \AAA). One part of the dichotomy follows from our constructions, the other is an unpublished result of Michael Herman asserting that if \a is Diophantine, then any area preserving diffeomorphism with rotation number \a on the boundary (on at least one of the boundaries in the case of \AAA) displays smooth invariant curves arbitrarily close to the boundary which clearly precludes ergodicity or even topological transitivity.Comment: To appear in annales de l'EN