From uniform to partial hyperbolocity
Amie Wilkinson Northwestern University

In this talk, I will survey the history of and some recent developments in the smooth ergodic theory of dynamical systems. The focus will be on the property of {\em ergodicity}, which is a statistical form of chaotic behavior observed in, and conjectured to hold for, many classical dynamical systems. The study of smooth ergodic theory has its origins in Boltzmann's Ergodic Hypothesis of the late 19th Century. As a response to Boltzmann's hypothesis, which was formulated in the context of Hamiltonian Mechanics, Birkhoff and von Neumann defined ergodicity in the 1930's and proved their foundational ergodic theorems.

In smooth ergodic theory, there are two well-studied phenomena associated to opposite long-term behaviors: hyperbolicity, which produces ergodicity, and Kolmogorov-Arnold-Moser (KAM) phenomena, which are often obstructions to ergodicity. Both hyperbolicity and KAM tori persist under smooth perturbations of the system, and their ergodic/non-ergodic properties are therefore stable.

Partially hyperbolic systems -- the central subject of this talk -- display a mixture of both hyperbolic and non-hyperbolic (such as KAM) dynamics. When two such opposite behaviors -- stable ergodicity and stable non-ergodicity -- are combined, which behavior prevails? I will discuss recent work that sheds light on the answer to this question.

Colloquium, Department of Mathematics, Rice University
September 4, 2008, HB 227 4-5 PM

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