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Hamiltonian loops from the ergodic point of view

Hamiltonian loops from the ergodic point of view 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the European Mathematical Society Springer Journals

Hamiltonian loops from the ergodic point of view

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Publisher
Springer Journals
Copyright
Copyright © 1999 by Springer-Verlag Berlin Heidelberg & EMS
Subject
Mathematics; Mathematics, general
ISSN
1435-9855
DOI
10.1007/PL00011161
Publisher site
See Article on Publisher Site

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.

Journal

Journal of the European Mathematical SocietySpringer Journals

Published: Jan 1, 1999

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