On quasi-ergodic hypothesis and Arnold diffusion for nearly integrable systems


Vadim Kaloshin

University of Maryland.


Thursday, April 9, 2015


Talk at 4:30 p.m. in E25-111

Tea at 4:00 p.m in E17-401


Abstract: The famous ergodic hypothesis claims that a typical Hamiltonian dynamics on a typical energy surface is ergodic, however, KAM theory disproves this. It establishes a persistent set of positive measure of invariant KAM tori. The (weaker) quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says that a typical Hamiltonian dynamics on a typical energy surface has a dense orbit. This question is wide open.

In early 60th Arnold constructed an example of instabilities for a nearly integrable Hamiltonian of dimension n>2 and conjectured that this is a generic phenomenon, nowadays, called Arnold diffusion. In the last two decades a variety of powerful techniques to attack this problem were developed. In particular, Mather discovered a large class of invariant sets and a delicate variational technique to shadow them. During the talk we present a progress on proving Arnold diffusion and quasi-ergodic hypothesis.

Here are arXiv links about this work:

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Posted: November 7, 2014    URL: