Morse estimates for translated points on unit tangent bundles
[Inégalités de Morse pour les points translatés des fibrés tangents unitaires]
Allais, Simon1
1 IRMA, Université de Strasbourg, 7 rue Rene Descartes, 67084 Strasbourg (France)
Annales de l'Institut Fourier, Online first, 20 p.

In this article, we study conjectures of Sandon on the minimal number of translated points of contactomorphisms in the special case of the unit tangent bundle of a Riemannian manifold. We restrict ourselves to contactomorphisms of a unit tangent bundle that lift diffeomorphisms of the base homotopic to the identity. We prove that there exist sequences (p n ,t n ) where p n is a translated point of time-shift t n with t n →+∞ for a large class of manifolds. In the case of Zoll–Riemannian manifolds, we also prove estimates relating the number of translated points to either the sum of the Betti numbers of the bundle under a generic assumption or its cuplength under a C 0 -closedness assumption.

Dans cet article, nous étudions des conjectures de Sandon concernant le nombre minimal de points translatés dans le cas particulier du fibré tangent unitaire d’une variété riemannienne. Nous restreignons aux contactomorphismes du fibré tangent unitaire relevant les difféomorphismes de la base homotopes à l’identité. Nous montrons qu’il existe des suites (p n ,t n ) où p n est un point translaté de temps de décalage t n avec t n →+∞ pour une grande classe de variétés. Dans le cas des variétés riemanniennes–Zoll, nous montrons aussi des inégalités entre le nombre de points translatés et la somme des nombres de Betti du fibré sous une hypothèse générique ou la cuplength sous une hypothèse de proximité C 0 .

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DOI : 10.5802/aif.3732
Classification : 53C22, 53D25, 57R17, 58E10
Keywords: Arnold conjecture, Chas–Sullivan product, geodesics, Lusternik–Schnirelmann theory, Morse theory, translated points of contactomorphisms, unit tangent bundles, Zoll metrics
Mots-clés : Conjecture d’Arnold, produit de Chas–Sullivan, géodésiques, théorie de Lusternik–Schnirelmann, théorie de Morse, points translatés de contactomorphismes, fibrés unitaires tangents, métriques de Zoll

Allais, Simon 1

1 IRMA, Université de Strasbourg, 7 rue Rene Descartes, 67084 Strasbourg (France) 
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Allais, Simon. Morse estimates for translated points on unit tangent bundles. Annales de l'Institut Fourier, Online first, 20 p.

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