Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y. A loop h:S1→G is called strictly ergodic if for some irrational number α the associated skew product map T:S1×Y→S1×Y defined by T(t,y)=(t+α,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.
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Received July 7, 1998 / final version received September 14, 1998
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Polterovich, L. Hamiltonian loops from the ergodic point of view . J. Eur. Math. Soc. 1, 87–107 (1999). https://doi.org/10.1007/PL00011161
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DOI: https://doi.org/10.1007/PL00011161