Hyperbolic entire functions with full hyperbolic dimension and approximation by Eremenko-Lyubich functions

TL;DR

This paper constructs a hyperbolic entire function with full hyperbolic dimension, surpassing known limitations, by developing new approximation techniques within the Eremenko-Lyubich class and analyzing their dynamical properties.

Contribution

It introduces a method to approximate entire functions with prescribed behavior near infinity, expanding the class of functions with full hyperbolic dimension and quasiconformal conjugacy.

Findings

Existence of hyperbolic entire functions with hyperbolic dimension two

New approximation method using Cauchy integrals for Eremenko-Lyubich class functions

Applications to transcendental dynamics and counterexamples

Abstract

We show that there exists a hyperbolic entire function of finite order of growth such that the hyperbolic dimension---that is, the Hausdorff dimension of the set of points in the Julia set of whose orbit is bounded---is equal to two. This is in contrast to the rational case, where the Julia set of a hyperbolic map must have Hausdorff dimension less than two, and to the case of all known explicit hyperbolic entire functions. In order to obtain this example, we prove a general result on constructing entire functions in the Eremenko-Lyubich class with prescribed behavior near infinity, using Cauchy integrals. This result significantly increases the class of functions that were previously known to be approximable in this manner. Furthermore, we show that the approximating functions are quasiconformally conjugate to their original models, which simplifies the construction of dynamical counterexamples. We also give some further applications of our results to transcendental dynamics.