Iterates of Blaschke products and Peano curves

TL;DR

This paper studies the boundary behavior of sums of iterates of finite Blaschke products, showing how they can approximate any complex number and under certain conditions, their images form sets with interior.

Contribution

It provides new results on the convergence and image properties of series involving iterates of Blaschke products, linking complex dynamics with boundary value problems.

Findings

Any complex number can be approximated by sums of iterates at some boundary point.

The image of the unit circle can have a non-empty interior under certain convergence conditions.

The proofs use inductive constructions exploiting the dynamics of Blaschke products.

Abstract

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation and let $f^{n}$ be its $n$-th iterate. Given a sequence $\{a_{n}\}$ of complex numbers consider $F= \sum a_n f^{n}$. If $\{a_n\}$ tends to $0$ but $\sum |a_n| = \infty$, we prove that for any complex number $w$ there exists a point $\xi$ in the unit circle such that $\sum a_{n}f^{n}(\xi)$ converges and its sum is $w$. If $\sum |a_n| < \infty$ and the convergence is slow enough in a certain precise sense, then the image of the unit circle by $F$ has a non empty interior. The proofs are based on inductive constructions which use the beautiful interplay between the dynamics of $f$ as a selfmapping of the unit circle and those as a selfmapping of the unit disc.