微分动力系统的若干问题探索

The study of topological and ergodic theory on dynamics beyond uniform hyperbolicity is the core research in modern dynamical systems. Our project is mainly concentrated on two parts: one is to study the chaotic properties of Birkhorff Irregular set and Lyapunov Irregular set; another is to consider one conjecture by Abdenur and Diaz: for generic diffeomorphisms, shadowing implies uniform hyperbolicity. We will use tools such as Pesin theory, Liao theory and its development, perturbation technique and approximation of Lyapunov exponents to open up our research. One aim of this project is to expand and enrich the topological and ergodic theory of dynamics beyond uniform hyperbolicity, and another is to search new discriminant rule for uniform hyperbolicity. And these will become theoretical basis for hot problems of modern differential dynamical systems.

一致双曲之外动力系统的拓扑和遍历性质的探索是当前微分动力系统的核心研究内容，本项目将主要围绕两个方面：一是探索Birkhorff非正则点集、Lyapunov非正则点集的混沌特征，二是考虑Abdenur-Diaz提出的通有条件下shadowing意味着一致双曲的猜测，结合Pesin理论、廖理论及其最新发展、扰动技术、Lyapuonv指数逼近等成果，展开一系列相关热点问题的研究。本项目的研究对拓展和丰富一致双曲之外动力系统的拓扑理论、遍历理论和寻找一致双曲新型判别法都有重要意义，并为当前微分动力系统领域的很多热点问题奠定理论基础。

本项目的研究工作主要围绕动力系统与遍历论领域的伪轨跟踪、拓扑熵、Birkhoff非正则点集与Lyapunov非正则点集的混沌特征、回复性及相关课题展开，取得的主要成果有：（1）将Birkhoff遍历的非正则点集与回复点集的各种层次联系起来，在一定条件下证明了它们各自都能达到与系统本身一致的复杂性，此成果适用于双曲、非一致双曲系统和符号系统等；（2）我们对具有渐进平均跟踪的动力系统，证明饱和集的存在性并用它来证明Birkhoff遍历的非正则点集合的存在及稠密性等；（3）对具有控制分解但光滑度仅要求C^1的微分流，重新构造了Pesin集合与Pesin块，并利用廖先生的准双曲理论实现了Pesin集合与Pesin块上的伪轨跟踪性质及相应的一些应用等学术成果。这些成果丰富和拓展了动力系统特别是一致双曲之外动力系统的拓扑理论和遍历理论，也对后续研究流的熵消失现象、SRB-like测度的重分形等课题有着重要启发和帮助。