[Finitude des mesures de Gibbs sur les variétés non compactes à courbure négative pincée]
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.
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Keywords: Gibbs measures, thermodynamic formalism, geodesic flow, geometrically infinite manifolds, Kac lemma
Mot clés : mesures de Gibbs, formalisme thermodynamique, flot géodésique, variétés géométriquement infinies, lemme de Kac
Pit, Vincent 1 ; Schapira, Barbara 2
CC-BY-ND 4.0
@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},
pages = {457--510},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {68},
number = {2},
year = {2018},
doi = {10.5802/aif.3167},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3167/}
}
TY - JOUR AU - Pit, Vincent AU - Schapira, Barbara TI - Finiteness of Gibbs measures on noncompact manifolds with pinched negative curvature JO - Annales de l'Institut Fourier PY - 2018 SP - 457 EP - 510 VL - 68 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3167/ DO - 10.5802/aif.3167 LA - en ID - AIF_2018__68_2_457_0 ER -
%0 Journal Article %A Pit, Vincent %A Schapira, Barbara %T Finiteness of Gibbs measures on noncompact manifolds with pinched negative curvature %J Annales de l'Institut Fourier %D 2018 %P 457-510 %V 68 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3167/ %R 10.5802/aif.3167 %G en %F 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/articles/10.5802/aif.3167/
[1] An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, 50, American Mathematical Society, Providence, RI, 1997, xii+284 pages | Zbl
[2] Exemples de surfaces hyperboliques de type divergent, de mesure de Sullivan associées finies mais non géométriquement finies (unpublished)
[3] On the mixing property for hyperbolic systems, Isr. J. Math., Volume 129 (2002), pp. 61-76 | DOI | Zbl
[4] Suspension flows over countable Markov shifts, J. Stat. Phys., Volume 124 (2006) no. 1, pp. 207-230 | DOI | MR | Zbl
[5] Geometrical finiteness with variable negative curvature, Duke Math. J., Volume 77 (1995) no. 1, pp. 229-274 | DOI | Zbl
[6] The ergodic theory of Axiom A flows, Invent. Math., Volume 29 (1975) no. 3, pp. 181-202 | DOI | Zbl
[7] Gibbs measures on negatively curved manifolds, J. Dyn. Control Syst., Volume 9 (2003) no. 1, pp. 89-101 | DOI | Zbl
[8] Séries de Poincaré des groupes géométriquement finis, Isr. J. Math., Volume 118 (2000), pp. 109-124 | DOI | Zbl
[9] Geodesic flows on negatively curved manifolds. I, Ann. Math., Volume 95 (1972), pp. 492-510 | DOI | MR | Zbl
[10] Sur les groupes hyperboliques d’après Mikhael Gromov (Ghys, Étienne; de la Harpe, Pierre, eds.), Progress in Mathematics, 83, Birkhäuser, Boston, MA, 1990, xii+285 pages (Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988) | Zbl
[11] Recurrence and transience for suspension flows, Isr. J. Math., Volume 209 (2015) no. 2, pp. 547-592 | DOI | Zbl
[12] Principe variationnel et groupes kleiniens, Duke Math. J., Volume 125 (2004) no. 1, pp. 15-44 | DOI | Zbl
[13] Prescribing the behaviour of geodesics in negative curvature, Geom. Topol., Volume 14 (2010) no. 1, pp. 277-392 | DOI | Zbl
[14] Equilibrium states in negative curvature, Astérisque, 373, Société Mathématique de France, 2015, viii+281 pages | MR | Zbl
[15] On the Patterson-Sullivan measure of some discrete group of isometries, Isr. J. Math., Volume 133 (2003), pp. 77-88 | DOI | Zbl
[16] Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr., Nouv. Sér., 95, Société Mathématique de France, 2003, vi+96 pages | MR | Zbl
[17] Thermodynamic Formalism for countable Markov shifts, Ergodic Theory Dyn. Syst., Volume 19 (1999) no. 6, pp. 1565-1593 | DOI | Zbl
[18] Thermodynamic Formalism for Null Recurrent Potentials, Isr. J. Math., Volume 121 (2001), pp. 285-311 | DOI | Zbl
[19] Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math., Volume 153 (1984) no. 3-4, pp. 259-277 | DOI | Zbl
[20] Mean least recurrence time, J. Lond. Math. Soc., Volume 36 (1961), pp. 382-384 | DOI | Zbl
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