On Moebius and conformal maps between boundaries of CAT(-1) spaces

We consider Moebius and conformal homeomorphisms $f : \partial X \to \partial Y$ 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 $S(f) : \partial ^2 X \to \mathbb{R}$, 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 $U \subset \partial X$ such that $S(f)$ vanishes on $\partial^2 U$. 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 $f : \partial X \to \partial Y$ if and only if there is a topological conjugacy of geodesic flows $ϕ: T^1 X \to T^1 Y$ 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 $\frac{1}{2}\log 2$-dense in $Y$. We prove that if $X,Y$ are in addition metric trees then $f$ extends to a surjective isometry. For $C^1$ conformal maps $f : \partial X \to \partial Y$ 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 $(1, \log 2 + 12||S(f)||_{\infty})$ quasi-isometry. We also obtain a dynamical classification of Moebius self-maps $f : \partial X \to \partial X$ into three types, elliptic, parabolic and hyperbolic.

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