Colloquium

Statistical properties of chaotic dynamical systems

by Huyi Hu, Visiting Assistant Professor,Department of Mathematics and Dynamical System Center,Penn State University

Abstract

This talk is about some statistical properties of chaoticdynamical systems, including the existence of physicallyrelevant invariant measures (called SRB measures), theirspeeds of correlation decay, and the Central Limit Theorem.

I will begin with an exposition on some results for Anosovsystems, which have often been used as models of chaos.The statistical properties are due to Sinai, Ruelle, andothers and are well known in the subject.

Recently, it has been conjectured that the statisticalproperties of Anosov systems are enjoyed by a much wider class of chaotic dynamical systems. After a review of these classical results, I will explain that these problems are more delicate than one might think. I will show, for example, that systems that fail to be Anosov at only one point may -- and sometimes do -- exhibit totally different long term behavior. SRB measures may cease to exist, and even when they exist, correlation decay may suddenly change from exponential to power law, and the Central Limit Theorem may fail.