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We show that there exists a hyperbolic entire function  f  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  f  whose orbit is bounded, is equal to 2. This is in contrast to the rational case, where the Julia set of a hyperbolic map must have Hausdorff dimension less than 2, 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  ℬ 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.

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