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

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.

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

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• Hausdorff dimensions of escaping sets of transcendental entire functions
• Hyperbolic entire functions with full hyperbolic dimension and approximation by Eremenko-Lyubich functions
• Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds
• The order conjecture fails in S
• The Schwarzian derivative and the Wiman-Valiron property

Cited by in corpus (10)

• Hyperbolic entire functions with full hyperbolic dimension and approximation by Eremenko-Lyubich functions
• The Hausdorff dimension of Julia sets of meromorphic functions in the Speiser class
• Eventual hyperbolic dimension of entire functions and Poincaré functions of polynomials
• Thermodynamic formalism and integral means spectrum of asymptotic tracts for transcendental entire functions
• Second order linear differential equations with a basis of solutions having only real zeros
• Hausdorff dimension of escaping sets of meromorphic functions
• Entire functions arising from trees
• A bouquet of pseudo-arcs
• The Hausdorff dimension of escaping sets of meromorphic functions in the Speiser class
• Holomorphic motions, natural families of entire maps, and multiplier-like objects for wandering domains