Speaker：Xueting Tian ( Fudan University )

Time：2026-5-23 17:00

Location：Conference Room S206 at Experiment Building at Haiyun Campus

Abstract:

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.