Geodesic
Hamiltonian loops from the ergodic point of view
Journal of the European Mathematical Society, Tome 1 (1999) no. 1, pp. 87-107.

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Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y. A loop h is called strictly ergodic if for some irrational number ! the associated skew product map T defined by T(t,y)=(t+!,h(t)y) is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of G can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply connected ones). Further, we find a restriction on the homotopy classes of smooth strictly ergodic loops in the framework of Hofer's bi-invariant geometry on G. Namely, we prove that their asymptotic Hofer's norm must vanish. This result provides a link between ergodic theory and symplectic topology.
DOI : 10.1007/pl00011161
Classification : 58-XX, 28-XX, 00-XX
Keywords: 
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     author = {Leonid Polterovich},
     title = {Hamiltonian loops from the ergodic point of view},
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Leonid Polterovich. Hamiltonian loops from the ergodic point of view. Journal of the European Mathematical Society, Tome 1 (1999) no. 1, pp. 87-107. doi : 10.1007/pl00011161. http://geodesic.mathdoc.fr/articles/10.1007/pl00011161/

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