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.