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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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A uniform contraction principle for bounded Apollonian embeddings
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by Loïc Dubois and Hans Henrik Rugh PDF
Conform. Geom. Dyn. 15 (2011), 64-71 Request permission


Let $\widehat {H}=H \cup \{\infty \}$ denote the standard one-point completion of a real Hilbert space $H$. Given any non-trivial proper subset $U\subset \widehat {H}$ one may define the so-called “Apollonian” metric $d_U$ on $U$. When $U\subset V \subset \widehat {H}$ are nested proper subsets we show that their associated Apollonian metrics satisfy the following uniform contraction principle: Let $\Delta =\mathrm {diam}_{V}(U) \in [0,+\infty ]$ be the diameter of the smaller subsets with respect to the large. Then for every $x,y\in U$ we have \[ d_V(x,y) \leq \tanh \frac {\Delta }{4} \ \ d_U(x,y) .\] In dimension one, this contraction principle was established by Birkhoff [Bir57] for the Hilbert metric of finite segments on ${{\mathbb R}\textrm {P}}^1$. In dimension two it was shown by Dubois in [Dub09] for subsets of the Riemann sphere $\widehat {\mathbb {C}}\sim \widehat {\mathbb {R}^2}$. It is new in the generality stated here.
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Additional Information
  • Loïc Dubois
  • Affiliation: Department of Mathematics and Statistics, University of Helsinki, Finland FI-00014
  • Email:
  • Hans Henrik Rugh
  • Affiliation: University of Cergy-Pontoise, CNRS UMR 8088, France.
  • Email:
  • Received by editor(s): February 19, 2011
  • Published electronically: June 28, 2011
  • Additional Notes: This research was partially funded by the European Research Council.
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Conform. Geom. Dyn. 15 (2011), 64-71
  • MSC (2010): Primary 30F45, 53A30; Secondary 47H09, 30C35
  • DOI:
  • MathSciNet review: 2833473