Hausdorff dimension of limit sets for projective Anosov representations

Olivier Glorieux; Daniel Monclair; Nicolas Tholozan

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]

Olivier Glorieux ¹ ; Daniel Monclair ² ; Nicolas Tholozan 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.

Résumé
Abstract

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.

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.

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
Mot clés : Exposant critique, dimension de Hausdorff, représentation Anosov, géométrie de Hilbert

Affiliations des auteurs :

Olivier Glorieux ¹ ; Daniel Monclair ² ; Nicolas Tholozan 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

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

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Olivier Glorieux; Daniel Monclair; Nicolas Tholozan. 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. https://jep.centre-mersenne.org/articles/10.5802/jep.241/

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