On hyperbolic dimension gap for entire functions

Polynomials and entire functions whose hyperbolic dimension is strictly smaller than the Hausdorff dimension of their Julia set are known to exist but in all these examples the latter dimension is maximal, i.e. equal to two. In this paper we show that there exist hyperbolic entire functions $f$ having Hausdorff dimension of the Julia set $\text{HD}({\text{J}}_f)<2$ and hyperbolic dimension $\text{HypDim}(f)<\text{HD}({\text{J}}_f)$.