Title: Dynamical behavior that are statistically trivial but topologically complicated in chaotic systems

Speaker:   Prof. Xueting Tian ( Fudan University )

Time: Tue, May.11 2021, PM: 2:00-3:00

Location: Tencent Conference

Meeting  Number: 352 157 165

Meeting Code: 654321

Inviter: Prof. Dongkui Ma

Abstract: 

    In a dynamical system, by known Poincare recurrence theorem, Birkhoff ergodic theorem and Oseledec multiplicative ergodic theorem, there exists a totally full measure set such that every point in this set is recurrent and its orbit enters in its neighborhood with positive lower density,  the set of its emprical measures of time average is a singleton corresponding to an ergodic measure, and the Lyapunov exponents at this point with respect to the derivative of a differential system or a cocycle exist. However, it has been found that the points without exitence of time average can carry full topological entropy and strong distributional chaos in various chaotic systems including symbolic systems, uniformly hyperbolic systems and some known non-uniformly hyperbolic systems such as Katok map, Mane examples etc. In this talk we will consider more different asymptotic behavior and show that they are usually statistically trivial but topologically complicated in various chaotic systems: (1) Lyapunov irregular points can carry full topological entropy and strong distributional chaos; (2) Points without SRB or SRB-like behavior can carry full topological entropy and strong distributional chaos; (3) Points with or without transitive behavior, recurrent behavior by using different frequency can form more than thirty different dynamical behavior, most of which are discovered to be statistically trivial but all carry strong topological complexity in the sense of full topological entropy and distributional chaos.