Parameter spaces and distortion sequences of entire functions with wandering domains

Sebastian van Strien, Gustavo Rodrigues Ferreira

Publication date: 2 July 2023

Abstract: The structural stability of holomorphic functions has been the subject of much research in the last fifty years. Due to various technicalities, however, most of that work has focused on so-called finite-type functions, implying that functions with wandering domains - another hot topic of research in complex dynamics - have (for the most part) not been addressed in this context. Given an entire function

f

with a simply connected wandering domain

U

, we construct an object called a distortion sequence that, under some hypotheses, moves analytically as

f

moves within appropriate parameter families. In order to "ground" our discussion, we consider -- given an entire function

f

-- the set

M_{f}

of entire functions quasiconformally equivalent to

f

. Generalising earlier results for the finite-type case, we show that

M_{f}

admits the structure of a complex manifold (of possibly infinite dimension).

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