Fatou's associates



Evdoridou, Vasiliki, Rempe, Lasse ORCID: 0000-0001-8032-8580 and Sixsmith, David J
(2020) Fatou's associates. Arnold Mathematical Journal 6 (2020), 459-493.

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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.

Item Type: Article
Additional Information: 32 pages, 6 figures. V4: Author accepted manuscript. To appear in Arnold Mathematical Journal (special volume dedicated to Prof. Mikhail Lyubich). A number of figures added from V1; general revision throughout; minor corrections of the proofs in Sections 8 and 9
Uncontrolled Keywords: math.DS, math.DS, math.CV, 37F10 (primary), 30D05, 30J05 (secondary)
Depositing User: Symplectic Admin
Date Deposited: 03 Jun 2020 08:57
Last Modified: 23 Jun 2021 11:10
DOI: 10.1007/s40598-020-00148-6
Open Access URL: https://rdcu.be/b9e4h
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3089321