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Conformal Geometry and Dynamics

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A uniform contraction principle for bounded Apollonian embeddings

Authors: Loïc Dubois and Hans Henrik Rugh
Journal: Conform. Geom. Dyn. 15 (2011), 64-71
MSC (2010): Primary 30F45, 53A30; Secondary 47H09, 30C35
Published electronically: June 28, 2011
MathSciNet review: 2833473
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Abstract: 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

Hans Henrik Rugh
Affiliation: University of Cergy-Pontoise, CNRS UMR 8088, France.

Received by editor(s): February 19, 2011
Published electronically: June 28, 2011
Additional Notes: This research was partially funded by the European Research Council.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.