2012/03/28 by Biswas, Kingshook
#51F99 #Dynamical Systems (math.DS) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1203.6212
We consider Moebius and conformal homeomorphisms f : ∂ X → ∂ Y between boundaries of CAT(-1) spaces X,Y equipped with visual metrics. A conformal map f induces a topological conjugacy of the geodesic flows of X and Y, which is flip-equivariant if f is Moebius. We define a function S(f) : ∂ 2 X → ℝ, the \it integrated Schwarzian of f, which measures the deviation of the topological conjugacy from being flip-equivariant, in particular vanishing if f is Moebius. Conversely if X,Y are simply connected complete manifolds with pinched negative sectional curvatures, then f is Moebius on any open set U ⊂ ∂ X such that S(f) vanishes on ∂2 U. Indeed we obtain an explicit formula for the cross-ratio distortion in terms of the integrated Schwarzian. For such manifolds, we show that there is a Moebius homeomorphism f : ∂ X → ∂ Y if and only if there is a topological conjugacy of geodesic flows ϕ: T1 X → T1 Y with a certain uniform continuity property along geodesics. We show that if X,Y are proper, geodesically complete CAT(-1) spaces then any Moebius homeomorphism f extends to a (1, log 2)-quasi-isometry with image (1)/(2)log 2-dense in Y. We prove that if X,Y are in addition metric trees then f extends to a surjective isometry. For C1 conformal maps f : ∂ X → ∂ Y with bounded integrated Schwarzian and with domain X a simply connected negatively curved manifold with a lower bound on sectional curvature, similar arguments show that f extends to a (1, log 2 + 12||S(f)||∞) quasi-isometry. We also obtain a dynamical classification of Moebius self-maps f : ∂ X → ∂ X into three types, elliptic, parabolic and hyperbolic.