ON INVARIANCE OF ORDER AND THE AREA PROPERTY FOR FINITE-TYPE ENTIRE FUNCTIONS



Epstein, Adam and Rempe-Gillen, Lasse ORCID: 0000-0001-8032-8580
(2015) ON INVARIANCE OF ORDER AND THE AREA PROPERTY FOR FINITE-TYPE ENTIRE FUNCTIONS. ANNALES ACADEMIAE SCIENTIARUM FENNICAE-MATHEMATICA, 40 (2). pp. 573-599.

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Abstract

Let f be an entire function that has only finitely many critical and asymptotic values. Up to topological equivalence, the function $f$ is determined by combinatorial information, more precisely by an infinite graph known as a "line-complex". In this note, we discuss the natural question whether the order of growth of an entire function is determined by this combinatorial information. The search for conditions that imply a positive answer to this question leads us to the "area property", which turns out to be related to many interesting and important questions in conformal dynamics and function theory. These include a conjecture of Eremenko and Lyubich, the measurable dynamics of entire functions, and pushforwards of quadratic differentials. We also discuss evidence that invariance of order and the area property fail in general.

Item Type: Article
Additional Information: 29 pages. v3. Final accepted manuscript, to appear in Ann. Acad. Sci. Fennicae. Several corrections and clarifications; new counterexample to Conjecture 1.6 included in Theorem 4.7. Previous versions: v2 - Substantial revisions throughout, including new section on quadratic differentials; v1 - original preprint version
Uncontrolled Keywords: Transcendental entire function, order conjecture, area property, topological equivalence, Poincare function, quadratic differential
Subjects: ?? QA ??
Depositing User: Symplectic Admin
Date Deposited: 06 Jul 2015 07:58
Last Modified: 18 Sep 2023 21:47
DOI: 10.5186/aasfm.2015.4034
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/2015680