Finiteness of Gibbs measures on noncompact manifolds with pinched negative curvature  [ Finitude des mesures de Gibbs sur les variétés non compactes à courbure négative pincée ]
Annales de l'Institut Fourier, Tome 68 (2018) no. 2, pp. 457-510.

Nous donnons plusieurs critères caractérisant la finitude des mesures de Gibbs pour le flot géodésique sur les variétés à courbure négative, analogues à ceux proposés par Sarig pour les sous-décalages sur des alphabets infinis. Ces critères effectifs devraient permettre de trouver davantage d’exemples de mesures de Gibbs finies. En application, nous retrouvons le critère de Dal’bo–Otal–Peigné sur la finitude de la mesure de Bowen–Margulis pour des variétés hyperboliques géométriquement finies, ainsi que les exemples de Peigné de variétés à courbure négative géométriquement infinies possédant une mesure de Bowen–Margulis finie.

We characterize the finiteness of Gibbs measures for geodesic flows on negatively curved manifolds by several criteria, analogous to those proposed by Sarig for symbolic dynamical systems over an infinite alphabet. These criteria should be useful in the future to find more examples with finite Gibbs measures. As an application, we recover Dal’bo–Otal–Peigné criterion of finiteness for the Bowen–Margulis measure on geometrically finite hyperbolic manifolds, as well as Peigné’s examples of geometrically infinite manifolds having a finite Bowen–Margulis measure.

Reçu le : 2016-10-10
Révisé le : 2017-05-05
Accepté le : 2017-06-15
Publié le : 2018-04-18
DOI : https://doi.org/10.5802/aif.3167
Classification : 37D40,  37D35,  28D20,  37A35,  37A40
Mots clés: mesures de Gibbs, formalisme thermodynamique, flot géodésique, variétés géométriquement infinies, lemme de Kac
@article{AIF_2018__68_2_457_0,
     author = {Pit, Vincent and Schapira, Barbara},
     title = {Finiteness of Gibbs measures on noncompact manifolds with pinched negative curvature},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {2},
     year = {2018},
     pages = {457-510},
     doi = {10.5802/aif.3167},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2018__68_2_457_0/}
}
Pit, Vincent; Schapira, Barbara. Finiteness of Gibbs measures on noncompact manifolds with pinched negative curvature. Annales de l'Institut Fourier, Tome 68 (2018) no. 2, pp. 457-510. doi : 10.5802/aif.3167. https://aif.centre-mersenne.org/item/AIF_2018__68_2_457_0/

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