Analytic mappings of the unit disk with bounded compression

TL;DR

This paper investigates a class of analytic self-maps of the unit disk characterized by uniformly bounded hyperbolic diameters of images of hyperbolic balls, providing multiple characterizations involving geometric and measure-theoretic properties.

Contribution

It introduces new characterizations of these maps through geodesic rays, Aleksandrov-Clark measures, zero sets, and critical sets, advancing understanding of their geometric structure.

Findings

Characterizations via geodesic rays

Connections with Aleksandrov-Clark measures

Descriptions of zero and critical sets

Abstract

In this paper, we study analytic self-maps of the unit disk for which the hyperbolic diameters of the images of hyperbolic balls of radius 1 are uniformly bounded below. We give several characterizations of such maps involving the behaviour along geodesic rays, Aleksandrov-Clark measures, zero sets and critical sets.