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

Abstract

Let f be a transcendental meromorphic function of finite order ρ for which the set of finite singularities of f -1 is bounded. Suppose that ∞ is not an asymptotic value and that there exists M ∈ ℕ such that the multiplicity of all poles, except possibly finitely many, is at most M. For R > 0 let I R(f) be the set of all z ∈ ℂ for which lim inf n→∞ |f n(z)| ≥ R as n → ∞. Here f n denotes the n-th iterate of f. Let I(f) be the set of all z ∈ ℂ such that |f n(z)| → ∞ as n → ∞; that is, I(f) = ∩ R>0 I R(f). Denote the Hausdorff dimension of a set A ⊂ ℂ by HD(A). It is shown that lim R→∞ HD(I R(f)) ≤ 2Mρ/(2 + Mρ). In particular, HD(I(f)) ≤ 2Mρ/(2 + Mρ). These estimates are best possible: for given ρ and M we construct a function f such that HD(I(f)) = 2Mρ/(2 + Mρ) and HD(IR(f)) > 2Mρ/(2 + Mρ) for all R > 0. If f is as above but of infinite order, then the area of I R(f) is zero. This result does not hold without a restriction on the multiplicity of the poles. © 2012 American Mathematical Society.