Cartan Spaces and Natural Foliations on the Cotangent Bundle

高秩阿贝尔群的 Cartan 作用及其分类

Cartan Actions of Higher Rank Abelian Groups and their Classification

本文研究紧流形上具有 Anosov 元素稠密集和 1 维粗 Lyapunov 叶状结构的全 Cartan 作用，通过引入动态定义拓扑群的新工具，将其完全分类为低维 Anosov 流、微分同胚和仿射作用的组合，验证了 Katok-Spatzier 猜想并应用于 Zimmer 纲领。

Rigidity properties of Anosov optical hypersurfaces

We consider an optical hypersurface $Σ$ in the cotangent bundle $τ:T^*M\to M$ of a closed manifold $M$ endowed with a twisted symplectic structure. We show that if the characteristic foliation of $Σ$ is Anosov, then a smooth 1-form $θ$ on $M$ is exact if and only $τ^*θ$ has zero integral over every closed characteristic of $Σ$. This result is derived from a related theorem about magnetic flows which generalizes our work in \cite{DP}. Other rigidity issues are also discussed.