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.
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.
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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
@article{AIF_2023__73_6_2605_0, author = {Gelfert, Katrin and Ruggiero, Rafael O.}, title = {Geodesic flows modeled by expansive flows: {Compact} surfaces without conjugate points and continuous {Green} bundles}, journal = {Annales de l'Institut Fourier}, pages = {2605--2649}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {73}, number = {6}, year = {2023}, doi = {10.5802/aif.3574}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3574/} }
TY - JOUR AU - Gelfert, Katrin AU - Ruggiero, Rafael O. TI - Geodesic flows modeled by expansive flows: Compact surfaces without conjugate points and continuous Green bundles JO - Annales de l'Institut Fourier PY - 2023 SP - 2605 EP - 2649 VL - 73 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3574/ DO - 10.5802/aif.3574 LA - en ID - AIF_2023__73_6_2605_0 ER -
%0 Journal Article %A Gelfert, Katrin %A Ruggiero, Rafael O. %T Geodesic flows modeled by expansive flows: Compact surfaces without conjugate points and continuous Green bundles %J Annales de l'Institut Fourier %D 2023 %P 2605-2649 %V 73 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3574/ %R 10.5802/aif.3574 %G en %F AIF_2023__73_6_2605_0
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, Volume 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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