Abundance of Different Types of Ergodic Measures and Orbits in Chaotic Systems

报告人：田学廷（复旦大学）

时  间：2026年5月23日17:00

地  点：海韵园实验楼S206

内容摘要:

In this talk, we first focus on the abundance of ergodic measures. We investigate the intermediate entropy property under the observation of continuous functions. On the one hand, this refines Katok's intermediate entropy property to derive a multifractal version. On the other hand, it enables us to obtain the intermediate properties of Hausdorff dimension, geometric pressure, unstable Hausdorff dimension, first return rate, Lyapunov exponents, and other related quantities. The systems under consideration include uniformly hyperbolic diffeomorphisms or flows, nonuniformly hyperbolic diffeomorphisms or singular hyperbolic flows, as well as various symbolic dynamical systems. In this process, we introduce and establish a "multi-horseshoe" entropy-dense property for these systems, which, combined with the well-known conditional variational principles, allows us to achieve our research goals. Recently, we have also adopted alternative methods to show that ergodic measures supported on minimal sets also possess the aforementioned intermediate properties. Furthermore, we introduce some progress regarding how the abundance of different types of invariant measures can induce the abundance of orbital structures in chaotic dynamical systems. Specifically, we use statistical ω-limit sets, classified by different positive densities of visiting time, to characterize various types of orbital behaviors, including different forms of recurrence and non-recurrence.

个人简介：

田学廷，复旦大学数学科学学院教授、博士生导师，主要研究领域为动力系统与遍历论。在微分动力系统的拓扑理论和遍历理论课题上取得进展，已在 Advances in Mathematics 、 Transactions of the American mathematical Society 、Journal of the Institute of Mathematics of Jussieu、 Ergodic Theory and Dynamical Systems、Mathematische Zeitschrift、Annales de l'Institut Henri Poincaré Probabilités et Statistiques、Journal of Differential Equations 等杂志上发表学术论文40余篇。

联系人：吴伟胜