Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville   rotation number on the boundary
  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 One part of the dichotomy follows from our constructions, the other is an unpublished result of Michael Herman asserting that if $\a$ is Diophantine, then any area preserving diffeomorphism with rotation number $\a$ on the boundary (on at least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. 
Bassam Fayad and Maria Saprykina