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Weakly mixing diffeomorphisms preserving a measurable Riemannian metric with prescribed Liouville rotation behavior

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

    Mathematics Subject Classification: Primary: 37A05; Secondary: 37C40, 57R50, 53C99.

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  • Figure 1.  The action of the map $g_{\varepsilon}$.

    Figure 2.  The action of the map $g_{a,b,\varepsilon}$.

    Figure 3.  The map $\psi_\mu$ has the useful property of rotating several small cuboids individually while being the identity outside of a neighborhood of them.

    Figure 4.  The map $\phi_n$ is constructed as concatenation of a stretch map $C_\lambda$, a rotation $\varphi$, the map $\psi_\mu$ mentioned before, and $C_\lambda^{-1}$ (the inverse of the stretch map). The map thus constructed has the very useful property of stretching a cuboid (illustrated here by the underlying grey rectangle) in one direction (similar to what a hyperbolic map would do), yet it is almost an isometry on all of the smaller cuboids (illustrated here by black squares with letters). In particular, a partition element $\hat{I} \in \eta_n$ (the leftmost grey rectangle) is mapped to a set that has size almost 1 in one of its coordinates.

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