Inner functions associated to lifts of transcendental entire functions

Abstract

Let $f$ be a transcendental entire function, $V$ be a simply connected Fatou component of $f,$ and $U$ be a Fatou component with $f(U)\subset V.$ There is a natural way to associate $f|_U$ to an inner function, namely a function $g_f:=\psi^{-1}\circ f\circ\varphi,$ where $\varphi:\mathbb{D}\to U$ and $\psi:\mathbb{D}\to V$ are Riemann maps. Inner functions have been used as a tool in the study of the iterates of transcendental entire, and more recently meromorphic, functions. However, there are only a few examples where associated inner functions have been calculated explicitly, with the case where $f$ has infinite degree in $U$ being the least well understood and more complicated. In this paper, we introduce a general method for calculating associated inner functions to a wide class of entire functions arising as `lifts'. In particular, if $f$ is a lift of a transcendental entire function $h,$ we show that an inner function associated to $f|_U$ can be obtained by relating it to an inner function associated to $h|_G,$ where $G$ is the Fatou component that lifts to $U.$ This result significantly generalises the main part of a theorem by Evdoridou, Rempe and Sixmith, and can be applied to several functions that have been studied so far. In both finite- and infinite-degree settings, the results hold for forward-invariant Fatou components as well as for wandering domains.