On the conformal gauge of a compact metric space

Matias Carrasco Piaggio

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 3, page 495-548
  • ISSN: 0012-9593

Abstract

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In this article we study the Ahlfors regular conformal gauge of a compact metric space ( X , d ) , and its conformal dimension dim A R ( X , d ) . Using a sequence of finite coverings of  ( X , d ) , we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all the metrics in the gauge. We show how to compute dim A R ( X , d ) using the critical exponent Q N associated to the combinatorial modulus.

How to cite

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Carrasco Piaggio, Matias. "On the conformal gauge of a compact metric space." Annales scientifiques de l'École Normale Supérieure 46.3 (2013): 495-548. <http://eudml.org/doc/272223>.

@article{CarrascoPiaggio2013,
abstract = {In this article we study the Ahlfors regular conformal gauge of a compact metric space $(X,d)$, and its conformal dimension $\dim _\{AR\}(X,d)$. Using a sequence of finite coverings of $(X,d)$, we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all the metrics in the gauge. We show how to compute $\dim _\{AR\}(X,d)$ using the critical exponent $Q_N$ associated to the combinatorial modulus.},
author = {Carrasco Piaggio, Matias},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Ahlfors regular; conformal gauge; conformal dimension; combinatorial modulus; Gromov-hyperbolic},
language = {eng},
number = {3},
pages = {495-548},
publisher = {Société mathématique de France},
title = {On the conformal gauge of a compact metric space},
url = {http://eudml.org/doc/272223},
volume = {46},
year = {2013},
}

TY - JOUR
AU - Carrasco Piaggio, Matias
TI - On the conformal gauge of a compact metric space
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 3
SP - 495
EP - 548
AB - In this article we study the Ahlfors regular conformal gauge of a compact metric space $(X,d)$, and its conformal dimension $\dim _{AR}(X,d)$. Using a sequence of finite coverings of $(X,d)$, we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all the metrics in the gauge. We show how to compute $\dim _{AR}(X,d)$ using the critical exponent $Q_N$ associated to the combinatorial modulus.
LA - eng
KW - Ahlfors regular; conformal gauge; conformal dimension; combinatorial modulus; Gromov-hyperbolic
UR - http://eudml.org/doc/272223
ER -

References

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