Alhamed, Mashael, Rempe, Lasse ORCID: 0000-0001-8032-8580 and Sixsmith, Dave
(2020)
Geometrically finite transcendental entire functions.
Journal of the London Mathematical Society.
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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 for more restrictive classes of entire functions.
Item Type: | Article |
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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: | 05 Jul 2021 13:44 |
Last Modified: | 18 Jan 2023 21:37 |
Related URLs: | |
URI: | https://livrepository.liverpool.ac.uk/id/eprint/3128592 |
Available Versions of this Item
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Geometrically finite transcendental entire functions. (deposited 03 Jun 2020 08:57)
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Geometrically finite transcendental entire functions. (deposited 22 Mar 2021 10:58)
- Geometrically finite transcendental entire functions. (deposited 05 Jul 2021 13:44) [Currently Displayed]
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Geometrically finite transcendental entire functions. (deposited 22 Mar 2021 10:58)