Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y. A loop his 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.
Cite this article
Leonid Polterovich, Hamiltonian loops from the ergodic point of view. J. Eur. Math. Soc. 1 (1999), no. 1, pp. 87–107DOI 10.1007/PL00011161