Contractive analytic self-mappings of the disc

TL;DR

This paper characterizes contractive analytic self-maps of the unit disc using Aleksandrov-Clark measures, inner-outer factorization, and boundary behavior, revealing new structural insights.

Contribution

It provides a novel description of contractive maps via Aleksandrov-Clark measures and boundary mixing properties, expanding understanding of their boundary behavior.

Findings

Contractive maps are characterized by their Aleksandrov-Clark measures.

Inner contractive functions exhibit a specific boundary mixing property.

New results on the boundary behavior of contractive inner functions.

Abstract

Analytic self-maps of the unit disc whose hyperbolic derivative is uniformly bounded by a constant smaller than one, are called contractive. We describe these maps in terms of their Aleksandrov-Clark measures and in terms of their inner-outer factorization. In addition, we show that contractive inner functions can be described in terms of a certain mixing property of its boundary values. We also present other results on the boundary behavior of contractive inner functions.