Iterates of Blaschke Products and Peano Curves

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 self-mapping of the unit circle and those as a self-mapping of the unit disk. © 2022 The Author(s). Published by Oxford University Press. All rights reserved.