Hausdorff dimension of limit sets for projective Anosov representations

Glorieux, Olivier; Monclair, Daniel; Tholozan, Nicolas

doi:10.5802/jep.241

Hausdorff dimension of limit sets for projective Anosov representations
[Dimension de Hausdorff d’ensembles limites pour les représentations projectivement Anosov]

Glorieux, Olivier ¹ ; Monclair, Daniel ² ; Tholozan, Nicolas ³

¹ Lycée Chaptal 45 Bd des Batignolles, 75008 Paris
² Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay F-91405 Orsay Cedex, France
³ CNRS, ÉNS-PSL 45 rue d’Ulm, 75005 Paris, France

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1157-1193

Abstract (VO)
Résumé

We study the relation between critical exponents and Hausdorff dimensions of limit sets for projective Anosov representations. We prove that the Hausdorff dimension of the symmetric limit set in $P (ℝ^{n}) \times P ({ℝ^{n}}^{*})$ is bounded between two critical exponents associated respectively to a highest weight and a simple root.

Nous étudions la relation entre les exposants critiques et les dimensions de Hausdorff des ensembles limites pour les représentations projectivement Anosov. Nous prouvons que la dimension de Hausdorff de l’ensemble limite symétrique dans $P (ℝ^{n}) \times P ({ℝ^{n}}^{*})$ est bornée par deux exposants critiques associés respectivement à un plus haut poids et à une racine simple.

Reçu le : 2019-07-01
Accepté le : 2023-04-13
Publié le : 2023-08-22

DOI : 10.5802/jep.241

Classification : 22E40, 37D40, 53A20, 20F67
Keywords: Critical exponent, Hausdorff dimension, Anosov representation, Hilbert geometry
Mots-clés : Exposant critique, dimension de Hausdorff, représentation Anosov, géométrie de Hilbert

Affiliations des auteurs :

Glorieux, Olivier ¹ ; Monclair, Daniel ² ; Tholozan, Nicolas ³

¹ Lycée Chaptal 45 Bd des Batignolles, 75008 Paris
² Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay F-91405 Orsay Cedex, France
³ CNRS, ÉNS-PSL 45 rue d’Ulm, 75005 Paris, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JEP_2023__10__1157_0,
     author = {Glorieux, Olivier and Monclair, Daniel and Tholozan, Nicolas},
     title = {Hausdorff dimension of limit sets for projective {Anosov} representations},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1157--1193},
     year = {2023},
     publisher = {Ecole polytechnique},
     volume = {10},
     doi = {10.5802/jep.241},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/jep.241/}
}

TY  - JOUR
AU  - Glorieux, Olivier
AU  - Monclair, Daniel
AU  - Tholozan, Nicolas
TI  - Hausdorff dimension of limit sets for projective Anosov representations
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2023
SP  - 1157
EP  - 1193
VL  - 10
PB  - Ecole polytechnique
UR  - https://www.numdam.org/articles/10.5802/jep.241/
DO  - 10.5802/jep.241
LA  - en
ID  - JEP_2023__10__1157_0
ER  -

%0 Journal Article
%A Glorieux, Olivier
%A Monclair, Daniel
%A Tholozan, Nicolas
%T Hausdorff dimension of limit sets for projective Anosov representations
%J Journal de l’École polytechnique — Mathématiques
%D 2023
%P 1157-1193
%V 10
%I Ecole polytechnique
%U https://www.numdam.org/articles/10.5802/jep.241/
%R 10.5802/jep.241
%G en
%F JEP_2023__10__1157_0

Glorieux, Olivier; Monclair, Daniel; Tholozan, Nicolas. Hausdorff dimension of limit sets for projective Anosov representations. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1157-1193. doi: 10.5802/jep.241

Bibliographie
Cité par

[AMS95] Abels, H.; Margulis, G. A.; Soĭfer, G. A. Semigroups containing proximal linear maps, Israel J. Math., Volume 91 (1995) no. 1-3, pp. 1-30 | DOI | MR | Zbl

[Ben97] Benoist, Yves Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal., Volume 7 (1997) no. 1, pp. 1-47 | DOI | Zbl

[Ben01] Benoist, Yves Convexes divisibles, C. R. Acad. Sci. Paris Sér. I Math., Volume 332 (2001) no. 5, pp. 387-390 | DOI | MR | Zbl

[CK02] Coornaert, M.; Knieper, G. Growth of conjugacy classes in Gromov hyperbolic groups, Geom. Funct. Anal., Volume 12 (2002) no. 3, pp. 464-478 | DOI | MR | Zbl

[CM14] Crampon, Mickaël; Marquis, Ludovic Finitude géométrique en géométrie de Hilbert, Ann. Inst. Fourier (Grenoble), Volume 64 (2014) no. 6, pp. 2299-2377 | DOI | Numdam | Zbl

[Coo93] Coornaert, Michel Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov, Pacific J. Math., Volume 159 (1993) no. 2, pp. 241-270 http://projecteuclid.org/euclid.pjm/1102634263 | DOI | MR | Zbl

[Cra09] Crampon, Mickaël Entropies of strictly convex projective manifolds, J. Modern Dyn., Volume 3 (2009) no. 4, pp. 511-547 | DOI | MR | Zbl

[Cra11] Crampon, Mickaël Dynamics and entropies of Hilbert metrics, Ph. D. Thesis, Université de Strasbourg; Ruhr-Universität, Bochum (2011) | MR

[CTT19] Collier, Brian; Tholozan, Nicolas; Toulisse, Jérémy The geometry of maximal representations of surface groups into ${SO}_{0} (2, n)$ , Duke Math. J., Volume 168 (2019) no. 15, pp. 2873-2949 | DOI | MR | Zbl

[DGK17] Danciger, Jeffrey; Guéritaud, François; Kassel, Fanny Convex cocompact actions in real projective geometry, 2017 to appear in Ann. Sci. École Norm. Sup. (4) | arXiv

[DGK18] Danciger, Jeffrey; Guéritaud, François; Kassel, Fanny Convex cocompactness in pseudo-Riemannian hyperbolic spaces, Geom. Dedicata, Volume 192 (2018), pp. 87-126 | DOI | MR | Zbl

[DK22] Dey, Subhadip; Kapovich, Michael Patterson-Sullivan theory for Anosov subgroups, Trans. Amer. Math. Soc., Volume 375 (2022) no. 12, pp. 8687-8737 | DOI | MR | Zbl

[DOP00] Dal’bo, Françoise; Otal, Jean-Pierre; Peigné, Marc Séries de Poincaré des groupes géométriquement finis, Israel J. Math., Volume 118 (2000), pp. 109-124 | DOI | Zbl

[Ebe96] Eberlein, Patrick B. Geometry of nonpositively curved manifolds, Chicago Lectures in Math., University of Chicago Press, Chicago, IL, 1996 | MR

[GGKW17] Guéritaud, François; Guichard, Olivier; Kassel, Fanny; Wienhard, Anna Anosov representations and proper actions, Geom. Topol., Volume 21 (2017) no. 1, pp. 485-584 | DOI | MR | Zbl

[GM21] Glorieux, Olivier; Monclair, Daniel Critical exponent and Hausdorff dimension in pseudo-Riemannian hyperbolic geometry, Internat. Math. Res. Notices (2021) no. 18, pp. 13661-13729 | DOI | MR | Zbl

[Gro87] Gromov, M. Hyperbolic groups, Essays in group theory (Math. Sci. Res. Inst. Publ.), Volume 8, Springer, New York, 1987, pp. 75-263 | DOI | MR | Zbl

[Gui19] Guichard, Olivier Groupes convexes-cocompacts en rang supérieur [d’après Labourie, Kapovich, Leeb, Porti,...], Séminaire Bourbaki, volume 2017/2018 (Astérisque), Volume 414, Société Mathématique de France, Paris, 2019, pp. 95-123 (Exp. no. 1138) | DOI | MR | Zbl

[GW12] Guichard, Olivier; Wienhard, Anna Anosov representations: domains of discontinuity and applications, Invent. Math., Volume 190 (2012) no. 2, pp. 357-438 | DOI | MR | Zbl

[Hel01] Helgason, Sigurdur Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Math., 34, American Mathematical Society, Providence, RI, 2001 | DOI

[KLP17] Kapovich, Michael; Leeb, Bernhard; Porti, Joan Anosov subgroups: dynamical and geometric characterizations, European J. Math., Volume 3 (2017) no. 4, pp. 808-898 | DOI | MR | Zbl

[KLP18] Kapovich, Michael; Leeb, Bernhard; Porti, Joan A Morse lemma for quasigeodesics in symmetric spaces and Euclidean buildings, Geom. Topol., Volume 22 (2018) no. 7, pp. 3827-3923 | DOI | MR | Zbl

[Lab06] Labourie, François Anosov flows, surface groups and curves in projective space, Invent. Math., Volume 165 (2006) no. 1, pp. 51-114 | DOI | MR | Zbl

[Lin04] Link, Gabriele Measures on the geometric limit set in higher rank symmetric spaces, Séminaire de Théorie Spectrale et Géométrie. Année 2003–2004, Volume 22, Univ. Grenoble I, Saint-Martin-d’Hères, 2004, pp. 59-69 | DOI | MR | Numdam | Zbl

[Mes07] Mess, Geoffrey Lorentz spacetimes of constant curvature, Geom. Dedicata, Volume 126 (2007), pp. 3-45 | DOI | MR | Zbl

[PS17] Potrie, Rafael; Sambarino, Andrés Eigenvalues and entropy of a Hitchin representation, Invent. Math., Volume 209 (2017) no. 3, pp. 885-925 | DOI | MR | Zbl

[PSW19] Pozzetti, Maria Beatrice; Sambarino, Andrés; Wienhard, Anna Anosov representations with Lipschitz limit set, 2019 | arXiv

[PSW21] Pozzetti, Maria Beatrice; Sambarino, Andrés; Wienhard, Anna Conformality for a robust class of non-conformal attractors, J. reine angew. Math., Volume 774 (2021), pp. 1-51 | DOI | MR | Zbl

[Qui02a] Quint, Jean-François Divergence exponentielle des sous-groupes discrets en rang supérieur, Comment. Math. Helv., Volume 77 (2002) no. 3, pp. 563-608 | DOI | MR | Zbl

[Qui02b] Quint, Jean-François Mesures de Patterson-Sullivan en rang supérieur, Geom. Funct. Anal., Volume 12 (2002) no. 4, pp. 776-809 | DOI | Zbl

[Rob03] Roblin, Thomas Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. France (N.S.), 95, Société Mathématique de France, Paris, 2003 | DOI | MR | Numdam

[Sam14] Sambarino, Andrés Quantitative properties of convex representations, Comment. Math. Helv., Volume 89 (2014) no. 2, pp. 443-488 | DOI | MR | Zbl

[Sul79] Sullivan, Dennis The density at infinity of a discrete group of hyperbolic motions, Publ. Math. Inst. Hautes Études Sci. (1979) no. 50, pp. 171-202 | MR | DOI | Numdam | Zbl

[Zim21] Zimmer, Andrew Projective Anosov representations, convex cocompact actions, and rigidity, J. Differential Geom., Volume 119 (2021) no. 3, pp. 513-586 | DOI | MR | Zbl

Cité par Sources :