no. 2
Geometry and entropies in a fixed conformal class on surfaces
[Sur la géométrie et les entropies dans une classes conforme fixée d’une surface]
Barthelmé, Thomas1 ; Erchenko, Alena2
1 Queen’s University Department of Mathematics and Statistics Kingston, ON (Canada)
2 Stony Brook University Mathematics Department Stony Brook, NY (USA)
Annales de l'Institut Fourier, Tome 71 (2021) no. 2, pp. 731-755.

Pour une surface à caractéristique d’Euler négative, nous considérons la classe des métriques à courbure négative, à volume fixé, et conformément équivalent à une métrique fixée. Pour cette classe de métriques, nous montrons que l’entropie métrique du flot géodésique est flexible, mais qu’à contrario, il y a des restrictions sur l’entropie topologique ainsi que la systole. Ce faisant, nous obtenons aussi un lemme du collier, une décomposition « épaisse-fine », et un théorème de précompacité pour cette classe de métrique. Nous étendons aussi certains de nos résultats au classes de métriques où la condition de courbure négative est remplacée par la condition de ne pas avoir de points focaux et d’avoir une certaine borne pour l’intégrale de la partie positive de la courbure de Gauss.

We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed total area in a fixed conformal class. Moreover, we obtain a collar lemma, a thick-thin decomposition, and precompactness for the considered class of metrics. Also, we extend some of the results to metrics of fixed total area in a fixed conformal class with no focal points and with some integral bounds on the positive part of the Gaussian curvature.

Reçu le :
Révisé le :
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DOI : 10.5802/aif.3410
Classification : 37D40, 53C20
Keywords: conformal class, topological entropy, metric entropy
Mot clés : classe conforme, entropie topologique, entropie métrique
Barthelmé, Thomas 1 ; Erchenko, Alena 2

1 Queen’s University Department of Mathematics and Statistics Kingston, ON (Canada)
2 Stony Brook University Mathematics Department Stony Brook, NY (USA)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Barthelmé, Thomas; Erchenko, Alena. Geometry and entropies in a fixed conformal class on surfaces. Annales de l'Institut Fourier, Tome 71 (2021) no. 2, pp. 731-755. doi : 10.5802/aif.3410. https://aif.centre-mersenne.org/articles/10.5802/aif.3410/

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