Inverse reduction inequalities for spectral numbers and applications

Viterbo, Claude

doi:10.48550/arxiv.2203.13172

Cited by 2 publications

(2 citation statements)

References 14 publications

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“…The content of Section 6 replaces the use of this conjecture in the proof of the main theorem. Added in revision: the conjecture has been recently proved, in particular for M = T n , by Shelukhin in [She22] (see also later proofs in [GV22,Vit22]). Also a number of papers using the present paper or its ideas are [MVZ12], [MZ11], [SV10], [Vit18], [Bis19], [Vit21].…”

Section: Sup(f (A))mentioning

confidence: 96%

Symplectic Homogenization

Viterbo¹

2022

Journal De L’École Polytechnique — Mathématiques

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Let H(q, p) be a Hamiltonian on T * T n . Under suitable assumptions on H, we show that the sequence (H k ) k⩾1 defined by H k (q, p) = H(kq, p) converges in the γ-topology-defined in [Vit92]-to an integrable continuous Hamiltonian H(p). This is extended to the case of nonautonomous Hamiltonians, and the more general setting in which only some of the variables are homogenized: we consider the sequence H(kx, y, q, p) and prove it has a γ-limit H(y, q, p), thus yielding an "effective Hamiltonian". The goal of this paper is to prove convergence of the above sequences, state the first properties of the homogenization operator, and give some applications to solutions of Hamilton-Jacobi equations, construction of quasi-states, etc. We also prove that when H is convex in p, the function H coincides with Mather's α function defined in [Mat91] and associated to the Legendre dual of H. This gives a new proof-in the torus case-of its symplectic invariance first discovered by P. Bernard in [Ber07].Résumé (Homogénéisation symplectique). -Soit H(q, p) un hamiltonien défini sur T * T n . Sous des hypothèses convenables, on montre que la suite (H k ) k⩾1 définie par H k (q, p) = H(kq, p) converge pour la topologie γ, définie dans [Vit92], vers un hamiltonien intégrable H(p). Ceci s'étend au cas de hamiltoniens non-autonomes, et au cas où seulement certaines variables sont homogénéisées : par exemple la suite H k (kx, y, px, py) qui dans ce cas aura une limite H(y, px, py), qui est un « hamiltonien effectif ». Le but de cet article est de démontrer la convergence de ces suites, ainsi que les premières propriétés de l'opérateur d'homogénéisation et d'en donner des applications aux solutions d'équations de Hamilton-Jacobi, aux quasi-états symplectiques, etc. On démontre aussi que lorsque H est convexe en p, la fonction H coïncide avec la fonction α de Mather (cf. [Mat91]) associée au dual de Legendre de H. Cela redémontre, dans le cas du tore, que cette fonction est symplectiquement invariante, comme l'avait démontré P. Bernard ([Ber07]) dans le cas général.

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Section: Sup(f (A))mentioning

confidence: 96%

Symplectic Homogenization

Viterbo¹

2022

Journal De L’École Polytechnique — Mathématiques

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“…(2) The Viterbo conjecture on the spectral norm, i.e. (ii) in Corollary 1, has recently been proven for Zoll symmetric spaces and so-called string-invertible spaces by Shelukhin [She18,She19], and for a large class of manifolds -which includes homogeneous spaces of compact Lie groups -by Viterbo [Vit22].…”

Section: Ia Precise Statementsmentioning

confidence: 99%

Hausdorff limits of submanifolds of symplectic and contact manifolds

Jean-Philippe¹

2022

Preprint

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We study sequences of immersions respecting bounds coming from Riemannian geometry and apply the ensuing results to the study of sequences of submanifolds of symplectic and contact manifolds. This allows us to study the subtle interaction between the Hausdorff metric and the Lagrangian Hofer and spectral metrics. In the process, we get proofs of metric versions of the nearby Lagrangian conjecture and of the Viterbo conjecture on the spectral norm. We also get C 0 -rigidity results in the vein of the Gromov-Eliashberg theorem for a vast class of important submanifolds of symplectic and contact manifoldseven when no Riemannian bounds are present.

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Inverse reduction inequalities for spectral numbers and applications

Cited by 2 publications

References 14 publications

Symplectic Homogenization

Symplectic Homogenization

Hausdorff limits of submanifolds of symplectic and contact manifolds

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