A uniform contraction principle for bounded Apollonian embeddings

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.