A uniform contraction principle for bounded Apollonian embeddings

Abstract

Let H ^ = H ∪ { ∞ } \widehat {H}=H \cup \{\infty \} denote the standard one-point completion of a real Hilbert space H H . Given any non-trivial proper subset U ⊂ H ^ U\subset \widehat {H} one may define the so-called “Apollonian” metric d U d_U on U U . When U ⊂ V ⊂ H ^ U\subset V \subset \widehat {H} are nested proper subsets we show that their associated Apollonian metrics satisfy the following uniform contraction principle: Let Δ = d i a m V ( U ) ∈ [ 0 , + ∞ ] \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 ∈ U x,y\in U we have \[ d V ( x , y ) ≤ tanh ⁡ Δ 4 d U ( x , y ) . d_V(x,y) \leq \tanh \frac {\Delta }{4} \ \ d_U(x,y) . \] In dimension one, this contraction principle was established by Birkhoff [Bir57] for the Hilbert metric of finite segments on R P 1 {{\mathbb R}\textrm {P}}^1 . In dimension two it was shown by Dubois in [Dub09] for subsets of the Riemann sphere C ^ ∼ R 2 ^ \widehat {\mathbb {C}}\sim \widehat {\mathbb {R}^2} . It is new in the generality stated here.