Minimal volume entropy for graphs
Author:
Seonhee Lim
Journal:
Trans. Amer. Math. Soc. 360 (2008), 5089-5100
MSC (2000):
Primary 37A35, 20E08
DOI:
https://doi.org/10.1090/S0002-9947-08-04227-X
Published electronically:
May 14, 2008
MathSciNet review:
2415065
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Among the normalized metrics on a graph, we show the existence and the uniqueness of an entropy-minimizing metric, and give explicit formulas for the minimal volume entropy and the metric realizing it.
Parmi les distances normalisées sur un graphe, nous montrons l’existence et l’unicité d’une distance qui minimise l’entropie, et nous donnons des formules explicites pour l’entropie volumique minimale et la distance qui la réalise.
- Hyman Bass, Covering theory for graphs of groups, J. Pure Appl. Algebra 89 (1993), no. 1-2, 3–47. MR 1239551, DOI https://doi.org/10.1016/0022-4049%2893%2990085-8
- G. Besson, G. Courtois, and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal. 5 (1995), no. 5, 731–799 (French). MR 1354289, DOI https://doi.org/10.1007/BF01897050
- Hyman Bass and Alexander Lubotzky, Tree lattices, Progress in Mathematics, vol. 176, Birkhäuser Boston, Inc., Boston, MA, 2001. With appendices by Bass, L. Carbone, Lubotzky, G. Rosenberg and J. Tits. MR 1794898
- Marc Bourdon, Structure conforme au bord et flot géodésique d’un ${\rm CAT}(-1)$-espace, Enseign. Math. (2) 41 (1995), no. 1-2, 63–102 (French, with English and French summaries). MR 1341941
- F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251 (French). MR 327923
- F. R. Gantmacher, Matrizenrechnung. II. Spezielle Fragen und Anwendungen, Hochschulbücher für Mathematik, Bd. 37, VEB Deutscher Verlag der Wissenschaften, Berlin, 1959 (German). MR 0107647
- Michael Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99 (1983). MR 686042
- Laurent Guillopé, Entropies et spectres, Osaka J. Math. 31 (1994), no. 2, 247–289 (French). MR 1296840
- Sa’ar Hersonsky and John Hubbard, Groups of automorphisms of trees and their limit sets, Ergodic Theory Dynam. Systems 17 (1997), no. 4, 869–884. MR 1468105, DOI https://doi.org/10.1017/S0143385797085040
- A. Katok, Entropy and closed geodesics, Ergodic Theory Dynam. Systems 2 (1982), no. 3-4, 339–365 (1983). MR 721728, DOI https://doi.org/10.1017/S0143385700001656
- I. Kapovich, T. Nagnibeda, The Patterson-Sullivan embedding and minimal volume entropy for outer space, preprint (http://arxiv.org/abs/math.GR/0504445), 2005.
- Seonhee Lim, Counting overlattices in automorphism groups of trees, Geom. Dedicata 118 (2006), 1–21. MR 2239446, DOI https://doi.org/10.1007/s10711-004-4196-7
- Anthony Manning, Topological entropy for geodesic flows, Ann. of Math. (2) 110 (1979), no. 3, 567–573. MR 554385, DOI https://doi.org/10.2307/1971239
- I. Rivin, Growth in free groups (and other stories), preprint (http://arxiv.org/ abs/math.CO/9911076), 1999.
- G. Robert, Entropie et graphes, Prépublication 182, ENS Lyon, 1996.
- Thomas Roblin, Sur la fonction orbitale des groupes discrets en courbure négative, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 1, 145–151 (French, with English and French summaries). MR 1881574
- Jean-Pierre Serre, Arbres, amalgames, ${\rm SL}_{2}$, Société Mathématique de France, Paris, 1977 (French). Avec un sommaire anglais; Rédigé avec la collaboration de Hyman Bass; Astérisque, No. 46. MR 0476875
Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37A35, 20E08
Retrieve articles in all journals with MSC (2000): 37A35, 20E08
Additional Information
Seonhee Lim
Affiliation:
Department of Mathematics, Yale University, New Haven, Connecticut 06520-8283 – and – ENS-Paris, UMR 8553 CNRS, 45 rue d’Ulm, 75230 Paris Cedex 05, France
Address at time of publication:
Department of Mathematics, Cornell University, 593 Malott Hall, Ithaca, New York 14853-4201
Email:
seonhee.lim@yale.edu, Seonhee.Lim@ens.fr, slim@math.cornell.edu
Received by editor(s):
June 26, 2005
Received by editor(s) in revised form:
December 3, 2005
Published electronically:
May 14, 2008
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.