On some completions of the space of hamiltonian maps

Vincent Humilière

Displaying similar documents to “On some completions of the space of hamiltonian maps”

Hofer’s metrics and boundary depth

Michael Usher (2013)

Annales scientifiques de l'École Normale Supérieure

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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 $M \times M$ ...

Robust transitivity in hamiltonian dynamics

Meysam Nassiri, Enrique R. Pujals (2012)

Annales scientifiques de l'École Normale Supérieure

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A goal of this work is to study the dynamics in the complement of KAM tori with focus on non-local robust transitivity. We introduce $C^{r}$ open sets ( $r = 1, 2, \dots, \infty$ ) of symplectic diffeomorphisms and Hamiltonian systems, exhibitingrobustly transitive sets. We show that the $C^{\infty}$ closure of such open sets contains a variety of systems, including so-called unstable integrable systems. In addition, the existence of ergodic measures with large support is obtained for all those systems. A main ingredient of the...

Problems remaining NP-complete for sparse or dense graphs

Ingo Schiermeyer (1995)

Discussiones Mathematicae Graph Theory

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For each fixed pair α,c > 0 let INDEPENDENT SET ( $m \leq c n^{α}$ ) and INDEPENDENT SET ( $m \geq (ⁿ ₂) - c n^{α}$ ) be the problem INDEPENDENT SET restricted to graphs on n vertices with $m \leq c n^{α}$ or $m \geq (ⁿ ₂) - c n^{α}$ edges, respectively. Analogously, HAMILTONIAN CIRCUIT ( $m \leq n + c n^{α}$ ) and HAMILTONIAN PATH ( $m \leq n + c n^{α}$ ) are the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH restricted to graphs with $m \leq n + c n^{α}$ edges. For each ϵ > 0 let HAMILTONIAN CIRCUIT (m ≥ (1 - ϵ)(ⁿ₂)) and HAMILTONIAN PATH (m ≥ (1 - ϵ)(ⁿ₂)) be the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH...

Special Lagrangian linear subspaces in product symplectic space

Małgorzata Mikosz (2004)

Banach Center Publications

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The notes consist of a study of special Lagrangian linear subspaces. We will give a condition for the graph of a linear symplectomorphism $f : (ℝ^{2 n}, σ = \sum_{i = 1}^{n} d x_{i} \land d y_{i}) \to (ℝ^{2 n}, σ)$ to be a special Lagrangian linear subspace in $(ℝ^{2 n} \times ℝ^{2 n}, ω = π * ₂ σ - π * ₁ σ)$ . This way a special symplectic subset in the symplectic group is introduced. A stratification of special Lagrangian Grassmannian $S Λ_{2 n} ≃ S U (2 n) / S O (2 n)$ is defined.

Generalized Conley-Zehnder index

Jean Gutt (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit $1$ as an eigenvalue. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this generalization is half integer valued. It is based on a Maslov-type index that they define for a continuous path of Lagrangians in a symplectic vector space $(W, \bar{Ω})$ , having chosen a given...

Inner and outer hamiltonian capacities

David Hermann (2004)

Bulletin de la Société Mathématique de France

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The aim of this paper is to compare two symplectic capacities in $ℂ^{n}$ related with periodic orbits of Hamiltonian systems: the Floer-Hofer capacity arising from symplectic homology, and the Viterbo capacity based on generating functions. It is shown here that the inner Floer-Hofer capacity is not larger than the Viterbo capacity and that they are equal for open sets with restricted contact type boundary. As an application, we prove that the Viterbo capacity of any compact Lagrangian submanifold...

Symplectic periods of the continuous spectrum of $GL (2 n)$

Shunsuke Yamana (2014)

Annales de l’institut Fourier

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We provide a formula for the symplectic period of an Eisenstein series on $GL (2 n)$ and determine when it is not identically zero.

Analytic invariants for the $1 : - 1$ resonance

José Pedro Gaivão (2013)

Annales de l’institut Fourier

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Associated to analytic Hamiltonian vector fields in $ℂ^{4}$ having an equilibrium point satisfying a non semisimple $1 : - 1$ resonance, we construct two constants that are invariant with respect to local analytic symplectic changes of coordinates. These invariants vanish when the Hamiltonian is integrable. We also prove that one of these invariants does not vanish on an open and dense set.

Characterization of diffeomorphisms that are symplectomorphisms

Stanisław Janeczko, Zbigniew Jelonek (2009)

Fundamenta Mathematicae

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Let $(X, ω_{X})$ and $(Y, ω_{Y})$ be compact symplectic manifolds (resp. symplectic manifolds) of dimension 2n > 2. Fix 0 < s < n (resp. 0 < k ≤ n) and assume that a diffeomorphism Φ : X → Y maps all 2s-dimensional symplectic submanifolds of X to symplectic submanifolds of Y (resp. all isotropic k-dimensional tori of X to isotropic tori of Y). We prove that in both cases Φ is a conformal symplectomorphism, i.e., there is a constant c ≠0 such that $Φ * ω_{Y} = c ω_{X}$ .

Hamiltonian-colored powers of strong digraphs

Garry Johns, Ryan Jones, Kyle Kolasinski, Ping Zhang (2012)

Discussiones Mathematicae Graph Theory

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For a strong oriented graph D of order n and diameter d and an integer k with 1 ≤ k ≤ d, the kth power $D^{k}$ of D is that digraph having vertex set V(D) with the property that (u, v) is an arc of $D^{k}$ if the directed distance $^{\to} d_{D} (u, v)$ from u to v in D is at most k. For every strong digraph D of order n ≥ 2 and every integer k ≥ ⌈n/2⌉, the digraph $D^{k}$ is Hamiltonian and the lower bound ⌈n/2⌉ is sharp. The digraph $D^{k}$ is distance-colored if each arc (u, v) of $D^{k}$ is assigned the color i where $i =^{\to} d_{D} (u, v)$ . The digraph...

Rigidity and gluing for Morse and Novikov complexes

Octav Cornea, Andrew Ranicki (2003)

Journal of the European Mathematical Society

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We obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closed symplectic manifold $(M, ω)$ with $c_{1} {|_{π_{2} (M)} = [ω] |}_{π_{2} (M)} = 0$ . The rigidity results for these complexes show that the complex of a fixed generic function/hamiltonian is a retract of the Morse (respectively Novikov or Floer) complex of any other sufficiently $C^{0}$ close generic...

The cubic Szegő equation

Patrick Gérard, Sandrine Grellier (2010)

Annales scientifiques de l'École Normale Supérieure

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We consider the following Hamiltonian equation on the $L^{2}$ Hardy space on the circle, $i \partial_{t} u = {Π (| u |}^{2} u),$ where $Π$ is the Szegő projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several...

$𝒞^{0}$ -rigidity of characteristics in symplectic geometry

Emmanuel Opshtein (2009)

Annales scientifiques de l'École Normale Supérieure

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The paper concerns a $𝒞^{0}$ -rigidity result for the characteristic foliations in symplectic geometry. A symplectic homeomorphism (in the sense of Eliashberg-Gromov) which preserves a smooth hypersurface also preserves its characteristic foliation.

Fredholm theory and transversality for the parametrized and for the $S^{1}$ -invariant symplectic action

Frédéric Bourgeois, Alexandru Oancea (2010)

Journal of the European Mathematical Society

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We study the parametrized Hamiltonian action functional for finite-dimensional families of Hamiltonians. We show that the linearized operator for the $L^{2}$ -gradient lines is Fredholm and surjective, for a generic choice of Hamiltonian and almost complex structure. We also establish the Fredholm property and transversality for generic $S^{1}$ -invariant families of Hamiltonians and almost complex structures, parametrized by odd-dimensional spheres. This is a foundational result used to define $S^{1}$ -equivariant...

Perturbation results for a class of singular Hamiltonian systems

Antonio Ambrosetti, Ivar Ekeland (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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The existence of solutions with prescribed period $T$ for a class of Hamiltonian systems with a Keplerian singularity is discussed.

Rabinowitz Floer homology and symplectic homology

Kai Cieliebak, Urs Frauenfelder, Alexandru Oancea (2010)

Annales scientifiques de l'École Normale Supérieure

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The first two authors have recently defined Rabinowitz Floer homology groups $R F H_{*} (M, W)$ associated to a separating exact embedding of a contact manifold $(M, ξ)$ into a symplectic manifold $(W, ω)$ . These depend only on the bounded component $V$ of $W ∖ M$ . We construct a long exact sequence in which symplectic cohomology of $V$ maps to symplectic homology of $V$ , which in turn maps to Rabinowitz Floer homology $R F H_{*} (M, W)$ , which then maps to symplectic cohomology of $V$ . We compute $R F H_{*} (S T^{*} L, T^{*} L)$ , where $S T^{*} L$ is the unit cosphere bundle of a closed...

Derivation of Hartree’s theory for mean-field Bose gases

Mathieu Lewin (2013)

Journées Équations aux dérivées partielles

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This article is a review of recent results with Phan Thành Nam, Nicolas Rougerie, Sylvia Serfaty and Jan Philip Solovej. We consider a system of $N$ bosons with an interaction of intensity $1 / N$ (mean-field regime). In the limit $N \to \infty$ , we prove that the first order in the expansion of the eigenvalues of the many-particle Hamiltonian is given by the nonlinear Hartree theory, whereas the next order is predicted by the Bogoliubov Hamiltonian. We also discuss the occurrence of Bose-Einstein condensation...

Factors and circuits in $K_{1, 3}$ -free graphs

Zdeněk Ryjáček (1989)

Banach Center Publications

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