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Quasi-morphismes et difféomorphismes hamiltoniens

Abstract : In this work, we study various invariants of algebraic and dynamical nature, defined on the group of Hamiltonian diffeomorphisms of a closed oriented surface. Occasionally we will also consider the group of Hamiltonian diffeomorphisms of certain symplectic manifolds of higher dimension. These invariants are constructed in the spirit of the classical Poincaré rotation number, or of the rotation vectors associated to diffeomorphisms of surfaces. Moreover all these invariants are related to the theory of bounded cohomology.

In the first chapter, we construct homogeneous quasi-morphisms on the group of Hamiltonian diffeomorphisms of a closed oriented surface of positive genus, which are homomorphisms when restricted to the subgroup of Hamiltonian diffeomorphisms supported in any open set diffeomorphic to a disc. These constructions are motivated by a question of Entov and Polterovich. In the second chapter, we construct a quasi-morphism on the universal cover of the group of Hamiltonian diffeomorphisms of a monotone symplectic manifold.

The third chapter contains some results concerning area preserving actions on surfaces of lattices in semisimple groups. In the spirit of the "Zimmer program" we show how the existence of many quasi-morphisms, combined with some vanishing result in bounded cohomology, might be useful to exclude the existence of actions of higher rank lattices. The last chapter contains some remarks around Hofer's metric on the group of Hamiltonian diffeomorphisms.
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https://tel.archives-ouvertes.fr/tel-00263607
Contributor : Pierre Py <>
Submitted on : Wednesday, March 12, 2008 - 3:39:45 PM
Last modification on : Tuesday, November 19, 2019 - 10:53:32 AM
Long-term archiving on: : Thursday, May 20, 2010 - 9:59:37 PM

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  • HAL Id : tel-00263607, version 1

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Pierre Py. Quasi-morphismes et difféomorphismes hamiltoniens. Mathématiques [math]. Ecole normale supérieure de lyon - ENS LYON, 2008. Français. ⟨tel-00263607⟩

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