Measure rigidity beyond uniform hyperbolicity: Invariant Measures for Cartan actions on Tori

Hyperbolicity and Rigidity for Fibred Partially Hyperbolic Systems

Every volume-preserving centre-bunched fibred partially hyperbolic system with 2-dimensional centre either (1) has two distinct centre Lyapunov exponents, or (2) exhibits an invariant continuous line field (or pair of line fields) tangent to the centre leaves, or (3) admits a continuous conformal structure on the centre leaves invariant under both the dynamics and the stable and unstable holonomies. The last two alternatives carry strong restrictions on the topology of the centre leaves: (2) can only occur on tori, and for (3) the centre leaves must be either tori or spheres. Moreover, under s…

Banach空间中部分正常双曲不变子 manifold 的不变子 manifold

Invariant manifolds of partially normally hyperbolic invariant manifolds in Banach spaces

本文研究了Banach空间中近似不变集在满足几何双曲性条件下的局部不变子 manifold 的存在性和正则性，提出了部分正常双曲性概念，扩展了正常双曲性和部分双曲性理论，并应用于分析带纹 tori 的动力学。

二次 Hopf 分岔中的正规双曲性

Normal Hyperbolicity in Secondary Hopf Bifurcations

本文结合已有文献证明二次 Hopf 分岔中出现的环面具有正规双曲性，并利用平均法为小时间周期扰动系统扩展相空间中正规双曲不变环面的分岔建立了充分条件。

基于伴随方法的周期轨道与准周期不变环面敏感性分析

Sensitivity Analysis for Periodic Orbits and Quasiperiodic Invariant Tori Using the Adjoint Method

本文提出基于伴随方法的框架，用于非线性约束解延拓及敏感性分析，并应用于周期轨道和准周期不变环面，证明伴随条件解的存在性，且示例表明附加敏感性分析成本极低。

Beltrami fields exhibit knots and chaos almost surely

In this paper we show that, with probability 1, a random Beltrami field exhibits chaotic regions that coexist with invariant tori of complicated topologies. The motivation to consider this question, which arises in the study of stationary Euler flows in dimension 3, is V.I. Arnold's 1965 conjecture that a typical Beltrami field exhibits the same complexity as the restriction to an energy hypersurface of a generic Hamiltonian system with two degrees of freedom. The proof hinges on the obtention of asymptotic bounds for the number of horseshoes, zeros, and knotted invariant tori and periodic tra…

Diffeomorphism group valued cocycles over higher rank abelian Anosov actions

We prove that every smooth diffeomorphism group valued cocycle over certain abelian Anosov actions on tori (and more generally on infranilmanifolds), is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialisation apply, via the existing global rigidity results, to maximal Cartan actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth…