[Dimension conforme et module combinatoire des espaces métriques compacts]
Lʼobjet principal de cette Note est lʼétude de la dimension conforme Ahlfors régulière () dʼun espace métrique compact X. Cʼest un invariant numérique de quasisymétrie, introduit par P. Pansu. Elle joue actuellement un rôle important en théorie géométrique des groupes et en dynamique conforme. On montre comment calculer à partir de modules combinatoires en considérant un exposant critique Q. Comme conséquence de lʼégalité , on obtient un critère général de dimension un. Les conditions sont données en termes de points de coupure locale de X. On donne par ailleurs des applications de ces résultats aux bords des groupes hyperboliques et aux ensembles de Julia des fractions rationnelles semi-hyperboliques.
In this Note we study the Ahlfors regular conformal dimension () of a compact metric space X. This is a quasisymmetric numerical invariant, introduced by P. Pansu. It plays nowadays an important role in geometric group theory and in conformal dynamics. We show how to compute using a critical exponent Q associated to the combinatorial modulus. As a consequence of the equality , we obtain a general criterion ensuring that the AR conformal dimension is 1. The conditions are stated in terms of local cut points of X. Finally, we give applications of these results to the boundaries of Gromov hyperbolic groups and to the Julia sets of semi-hyperbolic rational maps.
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@article{CRMATH_2012__350_3-4_141_0,
author = {Carrasco Piaggio, Matias},
title = {Conformal dimension and combinatorial modulus of compact metric spaces},
journal = {Comptes Rendus. Math\'ematique},
pages = {141--145},
publisher = {Elsevier},
volume = {350},
number = {3-4},
year = {2012},
doi = {10.1016/j.crma.2012.01.015},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2012.01.015/}
}
TY - JOUR AU - Carrasco Piaggio, Matias TI - Conformal dimension and combinatorial modulus of compact metric spaces JO - Comptes Rendus. Mathématique PY - 2012 SP - 141 EP - 145 VL - 350 IS - 3-4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2012.01.015/ DO - 10.1016/j.crma.2012.01.015 LA - en ID - CRMATH_2012__350_3-4_141_0 ER -
%0 Journal Article %A Carrasco Piaggio, Matias %T Conformal dimension and combinatorial modulus of compact metric spaces %J Comptes Rendus. Mathématique %D 2012 %P 141-145 %V 350 %N 3-4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2012.01.015/ %R 10.1016/j.crma.2012.01.015 %G en %F CRMATH_2012__350_3-4_141_0
Carrasco Piaggio, Matias. Conformal dimension and combinatorial modulus of compact metric spaces. Comptes Rendus. Mathématique, Tome 350 (2012) no. 3-4, pp. 141-145. doi : 10.1016/j.crma.2012.01.015. http://www.numdam.org/articles/10.1016/j.crma.2012.01.015/
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