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

Understand

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.