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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:
http://arxiv.org/abs/1212.1150
http://arxiv.org/abs/1412.7088
http://arxiv.org/abs/1410.1844
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