Analytic mappings of the unit disk which almost preserve hyperbolic area

TL;DR

This paper investigates analytic self-maps of the unit disk that nearly preserve hyperbolic area, providing characterizations through derivatives, distortion, critical points, and measures, and analyzing their Lyapunov exponents.

Contribution

It introduces new characterizations and analysis methods for maps that approximately preserve hyperbolic area, connecting geometric and measure-theoretic properties.

Findings

Characterizations involving angular derivatives and Lipschitz extensions

Relations between M"obius distortion and critical point distribution

Analysis of Lyapunov exponents of Aleksandrov-Clark measures

Abstract

In this paper, we study analytic self-maps of the unit disk which distort hyperbolic area of large hyperbolic disks by a bounded amount. We give a number of characterizations involving angular derivatives, Lipschitz extensions, M\"obius distortion, the distribution of critical points and Aleksandrov-Clark measures. We also study Lyapunov exponents of their Aleksandrov-Clark measures.