Minimal volume entropy for graphs
HTML articles powered by AMS MathViewer
- by Seonhee Lim PDF
- Trans. Amer. Math. Soc. 360 (2008), 5089-5100 Request permission
Abstract:
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.
References
- Hyman Bass, Covering theory for graphs of groups, J. Pure Appl. Algebra 89 (1993), no. 1-2, 3–47. MR 1239551, DOI 10.1016/0022-4049(93)90085-8
- 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 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, DOI 10.1007/978-1-4612-2098-5
- Marc Bourdon, Structure conforme au bord et flot géodésique d’un $\textrm {CAT}(-1)$-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, Band 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 10.1017/S0143385797085040
- A. Katok, Entropy and closed geodesics, Ergodic Theory Dynam. Systems 2 (1982), no. 3-4, 339–365 (1983). MR 721728, DOI 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 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 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, $\textrm {SL}_{2}$, Astérisque, No. 46, Société Mathématique de France, Paris, 1977 (French). Avec un sommaire anglais; Rédigé avec la collaboration de Hyman Bass. MR 0476875
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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2415065