Hamiltonian loops from the ergodic point of view |
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Authors: | Leonid Polterovich |
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Institution: | (1) School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel, e-mail: polterov@math.tau.ac.il, IL |
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Abstract: | 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.
Received July 7, 1998 / final version received September 14, 1998 |
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Keywords: | Mathematics Subject Classification (1991): 58Dxx 58F05 28D05 |
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