Geometrically finite transcendental entire functions

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

There is a more recent version of this item available.

	Text 2003.08884v2.pdf - Submitted version Download (1MB) \| Preview
	Text sigma-compactness-9.pdf - Author Accepted Manuscript Download (304kB) \| Preview

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 for more restrictive classes of entire functions.

Item Type:	Article
Additional Information:	41 pages, 3 figures. V3: Some expositional changes and clarifications in the proof of Proposition 7.1
Uncontrolled Keywords:	math.DS, math.DS, math.CV, Primary 37F10, Secondary 30D05, 30F45, 37F20
Divisions:	Faculty of Science and Engineering > School of Physical Sciences
Depositing User:	Symplectic Admin
Date Deposited:	22 Mar 2021 10:58
Last Modified:	18 Jan 2023 22:55
Related URLs:	Author
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3117796