Weakly mixing diffeomorphisms preserving a measurable Riemannian metric are dense in $\mathcal{A}_{\alpha}\left(M\right)$ for arbitrary Liouvillean number $\alpha$

Abstract

We show that on any smooth compact connected manifold of dimension $m\geq 2$ admitting a smooth non-trivial circle action $\mathcal{S} = \left\{S_t\right\}_{t \in \mathbb{R}}$, $S_{t+1}=S_t$, the set of weakly mixing $C^{\infty}$-diffeomorphisms which preserve both a smooth volume $\nu$ and a measurable Riemannian metric is dense in $\mathcal{A}_{\alpha} \left(M\right)= \overline{ \left\{h \circ S_{\alpha} \circ h^{-1} : h \in \text{Diff}^{\infty}\left(M, \nu\right) \right\}}^{C^{\infty}}$ for every Liouvillean number $\alpha$. The proof is based on a quantitative version of the Anosov-Katok-method with explicitly constructed conjugation maps and partitions.