Fatou's associates

Abstract

Suppose that $f$ is a transcendental entire function, $V \subsetneq \mathbb{C}$ is a simply connected domain, and $U$ is a connected component of $f^{-1}(V)$. Using Riemann maps, we associate the map $f \colon U \to V$ to an inner function $g \colon \mathbb{D} \to \mathbb{D}$. It is straightforward to see that $g$ is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of $f$ in $V$ lie away from the boundary, there is a strong relationship between singularities of $g$ and accesses to infinity in $U$. In the case where $U$ is a forward-invariant Fatou component of $f$, this leads to a very significant generalisation of earlier results on the number of singularities of the map $g$. If $U$ is a forward-invariant Fatou component of $f$ there are currently very few examples where the relationship between the pair $(f, U)$ and the function $g$ have been calculated. We study this relationship for several well-known families of transcendental entire functions. It is also natural to ask which finite Blaschke products can arise in this way, and we show the following: For every finite Blaschke product $g$ whose Julia set coincides with the unit circle, there exists a transcendental entire function $f$ with an invariant Fatou component such that $g$ is associated to $f$ in the above sense. Furthermore, there exists a single transcendental entire function $f$ with the property that any finite Blaschke product can be arbitrarily closely approximated by an inner function associated to the restriction of $f$ to a wandering domain.