Apollonian等距与Mbius变换

Apollonian Isometries and Mbius Transformations

设D是R^2中至少包含三个边界点的单连通区域，对任意x,y∈D，αD（x，y）表示D中关于x，y两点的Apollonian度量．1998年A．F．Beardon猜测：若．f：D→D是Apollonian等距映射，则f必是D上的Moebius变换．在该文中作者对D是圆的情况肯定并证明了A．F．Beardon的上述猜想。

Let D Cbelong to R^2 be a simply connected domain with boundary containing at least three points, and for any x, y E D, αD（x, y） denote the Apollonian metric in D with respect to x and y. A. F. Beardon gave the following conjecture in 1998： If f ： D→D be an Apollonian isometry, then f is a Moebius transformation. In this paper, the author affirms and proves the conjecture when D is a disk.