Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 573-599

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

Adam Epstein and Lasse Rempe-Gillen

University of Warwick, Mathematics Institute
Coventry CV4 7AL, United Kingdom; a.l.epstein 'at' warwick.ac.uk

University of Liverpool, Deptartment of Mathematical Sciences
Liverpool L69 7ZL, United Kingdom; l.rempe 'at' liverpool.ac.uk

Abstract. Let f : C → C 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.

2010 Mathematics Subject Classification: Primary 30D20; Secondary 30D05, 30D15, 30D35.

Key words: Transcendental entire function, order conjecture, area property, topological equivalence, Poincaré function, quadratic differential.

Reference to this article: A. Epstein and L. Rempe-Gillen: On invariance of order and the area property for finite-type entire functions. Ann. Acad. Sci. Fenn. Math. 40 (2015), 573-599.

Full document as PDF file

doi:10.5186/aasfm.2015.4034