# Hamiltonian loops from the ergodic point of view

Journal of the European Mathematical Society (1999)

- Volume: 001, Issue: 1, page 87-107
- ISSN: 1435-9855

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topPolterovich, Leonid. "Hamiltonian loops from the ergodic point of view." Journal of the European Mathematical Society 001.1 (1999): 87-107. <http://eudml.org/doc/277523>.

@article{Polterovich1999,

abstract = {Let $G$ be the group of Hamiltonian diffeomorphisms of a closed symplectic
manifold $Y$. A loop $h:S^1\rightarrow G$ is called strictly ergodic if for some irrational number the associated skew product map $T:S^1\times Y\rightarrow S^1\times Y$ defined by $T(t,y)=(t+\alpha ;h(t)y)$ 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.},

author = {Polterovich, Leonid},

journal = {Journal of the European Mathematical Society},

keywords = {Hamiltonian diffeomorphism; strictly ergodic loop; contractible strictly ergodic loop; asymptotic Hofer’s norm; Hofer’s bi-invariant geometry; Hamiltonian; ergodic; strictly ergodic; ergodic loops},

language = {eng},

number = {1},

pages = {87-107},

publisher = {European Mathematical Society Publishing House},

title = {Hamiltonian loops from the ergodic point of view},

url = {http://eudml.org/doc/277523},

volume = {001},

year = {1999},

}

TY - JOUR

AU - Polterovich, Leonid

TI - Hamiltonian loops from the ergodic point of view

JO - Journal of the European Mathematical Society

PY - 1999

PB - European Mathematical Society Publishing House

VL - 001

IS - 1

SP - 87

EP - 107

AB - Let $G$ be the group of Hamiltonian diffeomorphisms of a closed symplectic
manifold $Y$. A loop $h:S^1\rightarrow G$ is called strictly ergodic if for some irrational number the associated skew product map $T:S^1\times Y\rightarrow S^1\times Y$ defined by $T(t,y)=(t+\alpha ;h(t)y)$ 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.

LA - eng

KW - Hamiltonian diffeomorphism; strictly ergodic loop; contractible strictly ergodic loop; asymptotic Hofer’s norm; Hofer’s bi-invariant geometry; Hamiltonian; ergodic; strictly ergodic; ergodic loops

UR - http://eudml.org/doc/277523

ER -

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