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 $m\ge 2$ 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.

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