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Hamiltonian loops from the ergodic point of view

Leonid Polterovich (1999)

Journal of the European Mathematical Society

Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y . A loop h : S 1 G is called strictly ergodic if for some irrational number the associated skew product map T : S 1 × Y S 1 × 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...

Hamiltonian stability and subanalytic geometry

Laurent Niederman (2006)

Annales de l’institut Fourier

In the 70’s, Nekhorochev proved that for an analytic nearly integrable Hamiltonian system, the action variables of the unperturbed Hamiltonian remain nearly constant over an exponentially long time with respect to the size of the perturbation, provided that the unperturbed Hamiltonian satisfies some generic transversality condition known as steepness. Using theorems of real subanalytic geometry, we derive a geometric criterion for steepness: a numerical function h which is real analytic around a...

Herman’s last geometric theorem

Bassam Fayad, Raphaël Krikorian (2009)

Annales scientifiques de l'École Normale Supérieure

We present a proof of Herman’s Last Geometric Theorem asserting that if F is a smooth diffeomorphism of the annulus having the intersection property, then any given F -invariant smooth curve on which the rotation number of F is Diophantine is accumulated by a positive measure set of smooth invariant curves on which F is smoothly conjugated to rotation maps. This implies in particular that a Diophantine elliptic fixed point of an area preserving diffeomorphism of the plane is stable. The remarkable...

Hierarchy of integrable geodesic flows.

Peter Topalov (2000)

Publicacions Matemàtiques

A family of integrable geodesic flows is obtained. Any such a family corresponds to a pair of geodesically equivalent metrics.

Hofer’s metrics and boundary depth

Michael Usher (2013)

Annales scientifiques de l'École Normale Supérieure

We show that if ( M , ω ) is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer’s metric on the group of Hamiltonian diffeomorphisms of  ( M , ω ) has infinite diameter, and indeed admits infinite-dimensional quasi-isometrically embedded normed vector spaces. A similar conclusion applies to Hofer’s metric on various spaces of Lagrangian submanifolds, including those Hamiltonian-isotopic to the diagonal in  M × M when M satisfies...

Homoclinic orbits for a class of singular second order Hamiltonian systems in ℝ3

Joanna Janczewska, Jakub Maksymiuk (2012)

Open Mathematics

We consider a conservative second order Hamiltonian system q ¨ + V ( q ) = 0 in ℝ3 with a potential V having a global maximum at the origin and a line l ∩ 0 = ϑ as a set of singular points. Under a certain compactness condition on V at infinity and a strong force condition at singular points we study, by the use of variational methods and geometrical arguments, the existence of homoclinic solutions of the system.

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