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

TL;DR

This paper proves that on certain manifolds with circle actions, weakly mixing diffeomorphisms preserving volume and a measurable Riemannian metric are dense in a specific conjugacy class for Liouvillean rotation numbers, using a quantitative Anosov-Katok method.

Contribution

It establishes the density of weakly mixing volume-preserving diffeomorphisms with a measurable Riemannian metric in conjugacy classes for Liouvillean numbers, extending previous results with explicit constructions.

Findings

Weakly mixing diffeomorphisms are dense in the conjugacy class for Liouvillean numbers.

The proof employs a quantitative Anosov-Katok method with explicit conjugation maps.

The result applies to manifolds with a smooth non-trivial circle action.

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.