Geometrically finite transcendental entire functions

Abstract

For polynomials, local connectivity of Julia sets is a much-studied and important property. Indeed, when the Julia set of a polynomial of degree $d\ge 2$ is locally connected, the topological dynamics can be completely described as a quotient of a much simpler system: angle $d$-tupling on the circle. For a transcendental entire function, local connectivity is less significant, but we may still ask for a description of the topological dynamics as the quotient of a simpler system. To this end, we introduce the notion of "docile" functions: a transcendental entire function with bounded postsingular set is docile if it is the quotient of a suitable disjoint-type function. Moreover, we prove docility for the large class of geometrically finite transcendental entire functions with bounded criticality on the Julia set. This can be seen as an analogue of the local connectivity of Julia sets for geometrically finite polynomials, first proved by Douady and Hubbard, and extends previous work of the second author and of Mihaljević for more restrictive classes of entire functions.

arXiv comment41 pages, 3 figures. V3: Some expositional changes and clarifications in the proof of Proposition 7.1

References in corpus (16)

• Iteration of meromorphic functions
• Local connectivity of Julia sets: expository lectures
• Fast escaping points of entire functions
• Topological Dynamics of Exponential Maps on their Escaping Sets
• Rigidity of escaping dynamics for transcendental entire functions
• Brushing the hairs of transcendental entire functions
• Classification of Escaping Exponential Maps
• Arc-like continua, Julia sets of entire functions, and Eremenko's Conjecture
• On a question of Eremenko concerning escaping components of entire functions
• Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds
• Hyperbolic entire functions with bounded Fatou components
• Holomorphic Removability of Julia Sets
• Hyperbolic entire functions and the Eremenko-Lyubich class: Class $\mathcal{B}$ or not class $\mathcal{B}$?
• A landing theorem for dynamic rays of geometrically finite entire functions
• A landing theorem for entire functions with bounded post-singular sets
• On connected preimages of simply-connected domains under entire functions

Cited by in corpus (3)

• The escaping set in transcendental dynamics
• Spiders' webs in the Eremenko-Lyubich class
• Exponential iteration and Borel sets