no. 6
Geodesic flows modeled by expansive flows: Compact surfaces without conjugate points and continuous Green bundles
[Flots géodésiques modelisés par des flots expansifs : surfaces compactes sans points conjugués avec fibrés de Green continus]
Gelfert, Katrin1 ; Ruggiero, Rafael O.2
1 Instituto de Matemática Universidade Federal do Rio de Janeiro Cidade Universitária – Ilha do Fundão Rio de Janeiro 21945-909 (Brazil)
2 Departamento de Matemática, PUC-Rio Rua Marqués de São Vicente, 225, CEP 22451-900 Rio de Janeiro, RJ, (Brazil)
Annales de l'Institut Fourier, Tome 73 (2023) no. 6, pp. 2605-2649.

Nous considérons le flot géodésique d’une surface compacte sans points conjugués, de genre supérieur à un et de fibrés de Green continus. L’identification de chaque bande de géodésiques bi-asymptotiques induit une relation d’équivalence dans le fibré unitaire tangent. Nous montrons que son espace quotient porte la structure d’une variété compacte tridimensionnelle. Cette variété porte un flot continu défini canoniquement par la relation d’équivalence, le flot quotient. Ce flot est expansif, semi-conjugué au flot géodésique de la surface en préservant le paramétrage du flot géodésique, et muni d’une structure de produit locale. Une étape essentielle de la preuve de ces propriétés est l’étude de la régularité des feuilletages horosphériques, nous montrons qu’ils sont bien tangents aux sous-fibrés de Green. En tant qu’application, il est montré que le flot géodésique a une mesure unique d’entropie maximale.

We study the geodesic flow of a compact surface without conjugate points and genus greater than one and continuous Green bundles. Identifying each strip of bi-asymptotic geodesics induces an equivalence relation on the unit tangent bundle. Its quotient space is shown to carry the structure of a 3-dimensional compact manifold. This manifold carries a canonically defined continuous flow which is expansive, time-preserving semi-conjugate to the geodesic flow, and has a local product structure. An essential step towards the proof of these properties is to study regularity properties of the horospherical foliations and to show that they are indeed tangent to the Green subbundles. As an application it is shown that the geodesic flow has a unique measure of maximal entropy.

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Révisé le :
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DOI : 10.5802/aif.3574
Classification : 53D25, 37D40, 37D25, 37D35, 28D20, 28D99
Keywords: Geodesic flows, conjugate points, expansive flow, Green bundles, measure of maximal entropy.
Mot clés : Flots géodésiques, points conjugués, flot expansif, fibrés de Green, mesure d’entropie maximale.
Gelfert, Katrin 1 ; Ruggiero, Rafael O. 2

1 Instituto de Matemática Universidade Federal do Rio de Janeiro Cidade Universitária – Ilha do Fundão Rio de Janeiro 21945-909 (Brazil)
2 Departamento de Matemática, PUC-Rio Rua Marqués de São Vicente, 225, CEP 22451-900 Rio de Janeiro, RJ, (Brazil)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Gelfert, Katrin; Ruggiero, Rafael O. Geodesic flows modeled by expansive flows: Compact surfaces without conjugate points and continuous Green bundles. Annales de l'Institut Fourier, Tome 73 (2023) no. 6, pp. 2605-2649. doi : 10.5802/aif.3574. https://aif.centre-mersenne.org/articles/10.5802/aif.3574/

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