微分动力系统的同调分析

Dynamical systems on a compact manifold of finite dimension induce linear transformations on the homology groups of manifold. This project plans to analyze the iterations of simplexes by differentiable maps, and combine the global topological structure and the local entropy to study the relationship between the spectrum of linear transformations on homology groups and the topological entropy. This will be helpful for studing a longstanding open problem in the area of differentiable dynamical systems—Shub’s entropy conjecture: the spectral radii of linear transformations on homology groups are not bigger than the topological entropy. . We are working on the evolvement laws of systems corresponding to the spectrum of homology and entropy from three aspects: a wide range of expansivity, the invariant measures associated with homotopy classes and the approximation of differentiable dynamical systems.

有限维紧流形上的动力系统诱导了流形的同调群上的线性变换。本项目拟分析可微映射对单纯形的迭代，并把整体拓扑结构和局部熵结合起来研究同调群上线性变换的谱和拓扑熵的关系。这将有助于研究微分动力系统领域里一个长期的公开问题—Shub 熵猜测：同调群上线性变换的谱半径的对数不大于拓扑熵。. 我们将从三个方面来探讨同调谱和熵所对应的系统演化规律：大范围可扩性，相关于同伦类的不变测度以及微分动力系统的逼近。

一般而言，复杂系统通常展现混沌的现象。混沌的系统通常具有正熵，且熵越大，系统越复杂。如何估算熵，一直是动力系统研究中的一个困难而重要的问题，而熵猜想从实现性角度给出了研究机制。本项目围绕熵猜想研究中的局部可扩性(尾熵与观测尺度表达式)、不变测度产生机制(一般测度中心流形上熵的处理)、光滑性在演化中的控制作用(Yomdin-Gromov半代数几何理论的应用)，联系于几何拓扑性质，从局部和整体两个方面实现了熵的有效估计。