On the Hausdorff dimension of the escaping set of certain meromorphic functions

TL;DR

This paper investigates the Hausdorff dimension of the escaping set of certain transcendental meromorphic functions, establishing bounds based on growth order and pole multiplicities, with implications for the set’s measure.

Contribution

It provides a sharp bound for the Hausdorff dimension of the escaping set for functions in class B with finite order and bounded pole multiplicities, extending understanding of their fractal geometry.

Findings

Hausdorff dimension of I(f) is less than 2 under specified conditions

For infinite order functions, the escaping set has zero area

Bounded pole multiplicities are crucial for the results

Abstract

The escaping set I(f) of a transcendental meromorphic function f consists of all points which tend to infinity under iteration. The Eremenko-Lyubich class B consists of all transcendental meromorphic functions for which the set of finite critical and asymptotic values of f is bounded. It is shown that if f is in B and of finite order of growth, if infinity is not an asymptotic value of f and if the multiplicities of the poles of f are uniformly bounded, then the Hausdorff dimension of I(f) is strictly smaller than 2. In fact, we give a sharp bound for the Hausdorff dimension of I(f) in terms of the order of f and the bound for the multiplicities of the poles. If f satisfies the above hypotheses but is of infinite order, then the area of I(f) is zero. This result does not hold without a restriction on the multiplicities of the poles.