Contractive analytic self-mappings of the disc

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.