Inner functions associated to lifts of transcendental entire functions

Mathematics > Dynamical Systems [Submitted on 18 Jun 2026] Title:Inner functions associated to lifts of transcendental entire functions View PDF HTML (experimental)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. Current browse context: math.DS References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.