Convergence of linear combinations of iterates of an inner function

Abstract

Let $f$ be an inner function with $f(0)=0$ which is not a rotation and let $f^{n}$ be its $n$-th iterate. Let $\{a_{n}\}$ be a sequence of complex numbers. We prove that the series $\sum a_{n}f^{n}(\xi)$ converges at almost every point $\xi$ of the unit circle if and only if $\sum |a_n|^2 < \infty$. The main step in the proof is to show that under this assumption, the function $F= \sum a_n f^n$ has bounded mean oscillation. We also prove that $F$ is bounded on the unit disc if and only if $\sum |a_n| < \infty$. Finally we describe the sequences of coefficients $\{a_n \}$ such that $F$ belongs to other classical function spaces, as the disc algebra and the Dirichlet class.