Geodesic flow of nonstrictly convex Hilbert geometries
[Flot géodésique des géométries de Hilbert non strictement convexes]
Annales de l'Institut Fourier, Tome 70 (2020) no. 4, pp. 1563-1593.

Dans cet article, nous décrivons le comportement topologique du flot géodésique pour une classe de 3-variétés fermées réalisées sous forme de quotients de géométries de Hilbert non strictement convexes. La structure de ces 3-variétés est explicitement décrite par Benoist ; elles sont de Finsler avec des parties plates plongées de façon isométrique, mais hyperboliques loin des parties plates. Nous prouvons que le flot géodésique du quotient est topologiquement mélangeant et satisfait un lemme fermant d’Anosov non uniforme, avec applications au comptage d’entropie et d’orbites. Nous prouvons également l’expansivité de l’entropie pour le flot géodésique de tout quotient compact d’une géométrie de Hilbert, ce qui implique l’existence d’une mesure d’entropie maximale.

In this paper we describe the topological behavior of the geodesic flow for a class of closed 3-manifolds realized as quotients of nonstrictly convex Hilbert geometries. The structure of these 3-manifolds is described explicitly by Benoist; they are Finsler with isometrically embedded flats, but hyperbolic away from flats. We prove the geodesic flow of the quotient is topologically mixing and satisfies a nonuniform Anosov Closing Lemma, with applications to entropy and orbit counting. We also prove entropy-expansivity for the geodesic flow of any compact quotient of a Hilbert geometry, which implies existence of a measure of maximal entropy.

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DOI : https://doi.org/10.5802/aif.3358
Classification : 22E40,  37D40,  37D25,  37B40,  53A20
Mots clés : géométries de Hilbert, flot géodésique, non uniforme hyperbolicité, dynamique topologique
@article{AIF_2020__70_4_1563_0,
     author = {Bray, Harrison},
     title = {Geodesic flow of nonstrictly convex {Hilbert} geometries},
     journal = {Annales de l'Institut Fourier},
     pages = {1563--1593},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {70},
     number = {4},
     year = {2020},
     doi = {10.5802/aif.3358},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3358/}
}
Bray, Harrison. Geodesic flow of nonstrictly convex Hilbert geometries. Annales de l'Institut Fourier, Tome 70 (2020) no. 4, pp. 1563-1593. doi : 10.5802/aif.3358. https://aif.centre-mersenne.org/articles/10.5802/aif.3358/

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