Quasiergodic hypothesis and an almost dense trajectory on an energy surface

Lundi 7 décembre 2009 14:00-15:00 - V. kaloshin - Pennsylvania State University

Résumé : The classical Ergodic Hypothesis asks if a typical Hamiltonian flow on a typical energy surface is ergodic. KAM theory proved this not to be the case, as there are invariant sets of positive measure.
A Quasiegodic Hypothesis asks if a typical Hamiltonian flow on a typical energy surface is transitive, i.e. has a dense orbit. This question is still open. A different time Ehrenfest, Birkhoff, Arnold, Herman asked if there is an example of a Hamiltonian with a dense orbit on an energy surface. We give a partial answer.
We construct a nearly integral system of 3 degree of freedom such that it has an almost dense orbit. More exactly, it is dense in a set of almost maximal Lebesgue measure on a fixed energy surface. The proof relies on Mather variational method and the theory of normal forms. This is a joint work with Ke Zhang and Yong Zheng.

Lieu : bât. 425 - 113-115

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