Complutense University Library

Bounded distortion homeomorphisms on ultrametric spaces


Hughes, Bruce and Martínez Pérez , Álvaro and Morón, Manuel A. (2010) Bounded distortion homeomorphisms on ultrametric spaces. Annales academiae scientiarum fennicae-mathematica, 35 (2). pp. 473-492. ISSN 1239-629X

[img] PDF
Restringido a Repository staff only hasta 31 December 2020.


Official URL:


It is well-known that quasi-isometrics between R-trees induce power quasi-symmetric homeomorphisms between their ultrametric end spaces. This paper investigates power quasi-symmetric homeomorphisms between bounded, complete, uniformly perfect, ultrametric spaces (i.e., those ultrametric spaces arising up to similarity as the end spaces of bushy trees). A bounded distortion property is found that characterizes power quasi-symmetric homeomorphisms between such ultrametric spaces that are also pseudo-doubling. Moreover, examples are given showing the extent to which the power quasi-symmetry of homeomorphisms is not captured by the quasiconformal and bi-Holder conditions for this class of ultrametric spaces.

Item Type:Article
Uncontrolled Keywords:Tree; real tree; bushy tree; ultrametric; end space; quasi-isometry; quasiconformal; quasi-symmetric; PQ-symmetric; doubling metric space
Subjects:Sciences > Mathematics > Topology
ID Code:15030

[1] M. Bonk and O. Schramm. Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal., 10(2):266–306, 2000.

[2] Martin R. Bridson and André Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathemat- ical Sciences]. Springer-Verlag, Berlin, 1999.

[3] Sergei Buyalo and Viktor Schroeder. Elements of asymptotic geometry. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2007.

[4] V. Z. Fe˘ınberg. Compact ultrametric spaces. Dokl. Akad. Nauk SSSR, 214:1041–1044, 1974.

[5] É. Ghys and P. de la Harpe, editors. Sur les groupes hyperboliques d’après Mikhael Gromov, volume 83 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1990. Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988.

[6] Juha Heinonen. Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001.

[7] Bruce Hughes. Trees and ultrametric spaces: a categorical equivalence. Adv. Math., 189(1):148–191, 2004.

[8] Bruce Hughes. Trees, ultrametrics, and noncommutative geometry. Pure Appl. Math. Q., (to appear). arXiv:math/0605131v2 [math.OA].

[9] Á. Martínez-Pérez. Quasi-isometries between visual hyperbolic spaces. Preprint. arXiv:0810.4505 [math.GT].

[10] Á. Martínez-Pérez and Manuel A. Morón. Uniformly continuous maps between ends of Rtrees. Math. Z. DOI 10.1007/s00209-008-0431-5.

[11] Mozhgan Mirani. Classical trees and compact ultrametric spaces. Dissertation. Vanderbilt University, 2006.

[12] Mozhgan Mirani. Classical trees and compact ultrametric spaces. 2008. Preprint.

[13] Lee Mosher, Michah Sageev, and Kevin Whyte. Quasi-actions on trees. I. Bounded valence. Ann. of Math. (2), 158(1):115–164, 2003.

[14] Frédéric Paulin. Un groupe hyperbolique est déterminé par son bord. J. London Math. Soc. (2), 54(1):50–74, 1996.

[15] John Roe. Lectures on coarse geometry, volume 31 of University Lecture Series. American Mathematical Society, Providence, RI, 2003.

[16] Stephen Semmes. Metric spaces and mappings seen at many scales. Appendix in Metric Structures for Riemannian and Non-Riemannian Spaces by M. Gromov et al., Birkhäuser, 1999.

[17] P. Tukia and J. Väisälä. Quasisymmetric embeddings of metric spaces. Ann. Acad. Sci. Fenn. Ser. A I Math., 5(1):97–114, 1980.

Deposited On:26 Apr 2012 08:12
Last Modified:06 Feb 2014 10:14

Repository Staff Only: item control page