Almost Isometric Maps of the Hyperbolic Plane

Abstract

of D, then d(Tp,Tq) = d(p,q) for all TeJi thus maps in Ji are hyperbolic isometrics. The Schwarz-Pick theorem asserts that if/:Z)->Z) is analytic then / decreases distances, . fl .t . „ d{Ap\M)^d{p,q), (1.1) or lnnnitesimally, i-W/OI 0. Following C. McMullen, we write M(c) for the set of analytic / : / ) -> / ) such that whenever B is a hyperbolic ball in D, diam (/(£)) ^ diam(5)-c, where diam denotes diameter in the hyperbolic metric. For example, n M(C) = jt, c>0 while f(z) = z eM(c) provided c is large. This paper gives three characterizations of the set M(c). The first characterization concerns nearly isometric behavior along certain geodesies, and the second is in terms of angular derivatives at boundary points. Each/eM(c) is a Blaschke product, and the third characterization is by the distribution of the zeros. We thank Curt McMullen for bringing M(c) to our attention and for the results of the next section.