Hamiltonian loops from the ergodic point of view

Polterovich, 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 -