Hausdorff Dimension of Escaping Sets of Nevanlinna Functions

Abstract

We determine the exact values of Hausdorff dimensions of escaping sets of meromorphic functions with polynomial Schwarzian derivatives. This will follow from the relation between these functions and the second order differential equations in the complex plane.Mathematics Subject Classification: 30D05 (primary), 37F10 (secondary).

Hausdorff dimension of escaping sets of meromorphic functions II

“…Our starting point is the following theorem, which brings together results of several authors; see [GK18,McM87] (and also [Cui21b, Theorem 1]). By dim E we mean the Hausdorff dimension of the set E. Meromorphic functions in S 2 have explicit formulas; see Theorem 2.1 in the next section for a simple proof.…”

“…This set plays a fundamental role in recent studies of transcendental dynamics. Starting with McMullen [McM87], a wide range of research focuses on estimating the Hausdorff dimensions of escaping sets; see, for instance, [Bar08, BKS09, RS10, Sch07] for some entire functions and [BK12, Cui21b, GK18] for certain meromorphic functions. Some of these papers also treat special Speiser functions.…”

The Hausdorff dimension of Julia sets of meromorphic functions in the Speiser class

“…We have strict inequality in (1.1) for functions of the form R(e z ) [29] and for Nevanlinna functions [18]. In general, however, (1.1) is best possible.…”