一致双曲性之外的C^1微分同胚同宿类上的动力学与测度

For C^1-generic diffeomorphisms of smooth closed manifold, homoclinic classes are basic pieces of chain recurrent set, especially when periodic points are involved. Dynamics of diffeomorphisms can be described by expansiveness and Lyapunov exponents of probability measures that supported on homoclinic classes. Conversely, typical non-hyperbolic mechanisms such as homoclinic tangencies and heterodimensional cycles implies special kind of measures supported on homoclinic classes. It is thus desirable to investigate the relation between measures of homoclinic classes and the behaviors of homoclinic orbits. We propose to consider the following inter-related topics: 1) By perturbing a homoclinic tangency, to construct non-hyperbolic measures inside homoclinic classes for diffeomorphisms in an open and dense subset or a residual subset; 2) To characterize the differential structure (uniform hyperbolicity, partial hyperbolicity and dominated splittings) from the measure-theoretic viewpoint. The special research topics include the construction of non-hyperbolic measures, the estimation of the hyperbolicity of invariant sets along homoclinic orbits, and bifurcations that keep homoclinic relations. Geometric properties of dynamics (behaviors of invariant subsets and invariant manifolds) are analyzed and quantities (such as eigenvalues and Lyapunov exponents) are estimated, in order to achieve better understanding of dynamics of homoclinic classes beyond uniform hyperbolicity.

对于光滑闭流形上C^1通有的微分同胚，同宿类是含有周期点的回归集的基本单元。支撑在同宿类上的概率测度可以通过扩张性、非一致双曲性等不同角度刻画微分同胚本身的动力学性质。反之，以同宿切和异维环为代表的非双曲结构也能诱导出同宿类上满足特定性质的测度。因此，非常有必要考察轨道的动力学行为与同宿类上测度的相互影响。我们将研究如下问题： 1) 由扰动同宿切出发，在剩余集或稠密开集上同宿类内部非双曲测度的构造以及性质；2) 从测度论角度对同宿类微分结构（一致双曲，部分双曲，控制分解）的刻画。具体研究包括非双曲测度的构造、对同宿切点轨道附近的不变集双曲性的估计、保持同宿相关关系的分支构造。研究特色是通过对具体的动力学几何性质（例如不变集和不变流形）的分析以及对刻画量（例如特征值和Lyapunov指数）的估计，从测度观点揭示出一致双曲之外的同宿类的动力学性态。