Hausdorff dimension of radial and escaping points for transcendental meromorphic functions

Abstract

We consider a class of transcendental meromorphic functions $f:\mathbb{C}\mapsto\overline{\mathbb{C}}$ with infinitely many poles. Under some regularity assumption on the location of poles and the behavior of the function near the poles, we provide explicite lower bounds for the hyperbolic dimension (Hausdorff dimension of radial points) of the Julia set and upper bounds for the Hausdorff dimension of the set of escaping points in the Julia set. In particular, the Hausdorff dimension of the latter set is less than the Hausdorff dimension of the former set. Consequently, the Hausdorff dimension of the set of escaping points is less than 2, and the area of this set is equal to zero. The functions under consideration may have infinitely many singular values, and we do not even assume them to belong to the class $\mathcal{B}$. We only require the distance between the set of poles and the set of finite singular values to be positive.