On invariance of order and the area property for finite-type entire functions

Let f : C → C be an entire function that has only finitely many critical and asymptotic values. Up totopological 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 push forwards of quadratic differentials. We also discuss evidence that invariance of order and the area property fail in general