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- W2800633685 abstract "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 $phi : T^1 X to T^1 Y$ 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." @default.
- W2800633685 created "2018-05-17" @default.
- W2800633685 creator A5010133670 @default.
- W2800633685 date "2012-03-28" @default.
- W2800633685 modified "2023-09-24" @default.
- W2800633685 title "On Isometric Extension of Moebius Maps" @default.
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- W2800633685 hasPublicationYear "2012" @default.
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