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

Letf:C→Cbe an entire function that has only finitely many critical andasymptotic values. Up totopological equivalence, the functionfis determined by combinatorialinformation, more precisely by an infinite graph known as aline-complex. In this note, we discussthe natural question whether the order of growth of an entirefunction is determined by this combi-natorial information. The search for conditions that implya positive answer to this question leadsus to thearea property, which turns out to be related to many interesting and important questionsin 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 alsodiscuss evidence that invariance of order and the area property fail in general