Almost isometric maps of the hyperbolic disc

Abstract

Our main goal is to understand the paper [GP91] of J. Garnett and M. Papadimitrakis on almost isometries. An analytic self-map of the unit disc is called an almost isometry if there exists $c > 0$ such that $\operatorname{diam}(f(B)) \ge \operatorname{diam}(B) - c$ for any hyperbolic disc $B \subset \mathbb{D}$. Here, $\operatorname{diam}(E)$ means the hyperbolic diameter of $E$.

Three equivalent characterizations of almost isometries will be considered: one based on geodesics, one involving angular derivatives, and another one using the distribution of zeros. These results combine notions from hyperbolic geometry, complex analysis, and measure theory.

The project reveals that almost isometries form a rich subclass of Blaschke products. Furthermore, an explicit example of an almost isometry whose angular derivative diverges almost everywhere is constructed, demonstrating the subtle boundary behaviour of these functions.