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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

$\displaystyle d_V(x,y) \leq \tanh \frac{\Delta}{4} \ d_U(x,y) .$

In dimension one, this contraction principle was established by Birkhoff [Bir57] for the Hilbert metric of finite segments on $ {{\mathbb{R}}{\rm 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.