拓扑动力系统中熵和emergence理论的研究

关于熵的变分原理的推广与应用已成为动力系统的重要课题；emergence是近期引入的用于刻画系统遍历性质的概念，对微分动力系统中Newhouse现象的研究有重要作用。本项目利用熵和emergence理论对拓扑动力系统的轨道渐进表现及统计规律进行刻画，所用方法和技巧涉及到遍历理论、维数理论、几何测度论等领域。具体的，我们将研究以下三个问题：(1)对满足specification-like性质的动力系统，建立饱和集的packing熵变分原理，并将其应用于平均Li-Yorke混沌对集合的下界估计；对流的通有点等集合建立熵的变分原理；(2)基于emergence理论，我们拟对诱导动力系统引入entropy order，探究其与拓扑熵之间的关系；(3)从熵与维数的角度讨论高点态emergence集合的大小。预期成果将丰富拓扑动力系统的维数理论，并加深人们对诱导动力系统复杂性的认识。

The extension and application of the variational principle of entropy has been a crucial subject of dynamical systems; the concept of emergence was introduced recently in order to describe the ergodic property of systems, which plays an important role in the study of Newhouse phenomenon in differential dynamical systems. This project is to characterize the asymptotic behavior of orbits and statistical laws of topological dynamical systems by using entropy and emergence theory, the methods and techniques used in this project are involved in Ergodic theory, Dimension theory and Geometric measure theory etc. We intend to investigate the following three problems: (1)establish the variational principle of packing entropy of the saturated sets for systems with specification-like property, and apply it to the lower bound of packing entropy of the set of mean Li-Yorke pairs; establish the variational principle of entropy of generic sets for flows; (2)based on the emergence theory, we intend to introduce the notation of entropy order to induced dynamical systems, and explore its relationship with topological entropy; (3)discuss the size of the set of high pointwise emergence from the viewpoints of entropy and dimension. The expected results will enrich the dimension theory of topological dynamical systems and deepen people's understanding of the complexity of induced dynamic systems.