Geometrically finite transcendental entire functions



Alhamed, Mashael, Rempe-Gillen, Lasse ORCID: 0000-0001-8032-8580 and Sixsmith, Dave
Geometrically finite transcendental entire functions.

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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 $d\geq 2$ 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 Mihaljevi\'c-Brandt for more restrictive classes of entire functions.

Item Type: Article
Additional Information: 41 pages, 3 figures. V2: Minor changes and revision, some changes in notation in Sections 2 and 9
Uncontrolled Keywords: math.DS, math.DS, math.CV, Primary 37F10, Secondary 30D05, 30F45, 37F20
Depositing User: Symplectic Admin
Date Deposited: 03 Jun 2020 08:57
Last Modified: 03 Jun 2020 08:57
Related URLs:
URI: http://livrepository.liverpool.ac.uk/id/eprint/3089322