Fatou components and singularities of meromorphic functions

Abstract

We prove several results concerning the relative position of points in the postsingular set $P(f)$ of a meromorphic map $f$ and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer a question of Mihaljević-Brandt and Rempe-Gillen. For wandering domains we show that if the iterates $U_n$ of such a domain have uniformly bounded diameter, then there exists a sequence of postsingular values $p_n$ such that $\mathrm{dist}(p_n,\partial U_n)\to 0$ as $n\to \infty$. We also prove that if $U_n\cap P(f)=\varnothing$ and the postsingular set of $f$ lies at a positive distance from the Julia set (in $\mathbb{C}$) then any sequence of iterates of wandering domains must contain arbitrarily large disks. This allows to exclude the existence of wandering domains for some meromorphic maps with infinitely many poles and unbounded set of singular values.

19 pages, 3 figures

References in corpus (2)

• Accesses to infinity from Fatou components
• Absorbing sets and Baker domains for holomorphic maps

Cited by in corpus (5)

• Stable components in the parameter plane of transcendental functions of finite type
• Singular orbits and Baker domains
• Oscillating Wandering Domains for Functions with Escaping Singular Values
• Dynamics of projectable functions: Towards an atlas of wandering domains for a family of Newton maps
• A new family of entire functions with no wandering domains