# American Institute of Mathematical Sciences

April  2018, 38(4): 1615-1655. doi: 10.3934/dcds.2018067

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

 1 University of Education Vorarlberg, Liechtensteinerstrasse 33 - 37, 6800 Feldkirch, Austria 2 University of Hamburg, Department of Mathematics, Bundesstrasse 55, 20146 Hamburg, Germany

Received  November 2015 Revised  October 2017 Published  January 2018

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.

Citation: Roland Gunesch, Philipp Kunde. Weakly mixing diffeomorphisms preserving a measurable Riemannian metric with prescribed Liouville rotation behavior. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1615-1655. doi: 10.3934/dcds.2018067
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##### References:
The action of the map $g_{\varepsilon}$.
The action of the map $g_{a,b,\varepsilon}$.
The map $\psi_\mu$ has the useful property of rotating several small cuboids individually while being the identity outside of a neighborhood of them.
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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