On the Hausdorff dimension of the escaping set of certain meromorphic functions (Q2841405)

The paper concerns geometric properties of the escaping set of meromorphic functions of finite type. Let \(f\) be a transcendental meromorphic function of finite order \(\rho\) such that the set of finite singularities of \(f^{-1}\) is bounded. Suppose that \(\infty\) is not an asymptotic value and that there exists \(M \in \mathbb{N}\) 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 \(z \in \mathbb{C}\) for which \(\liminf_{n\to \infty}|f^n(z)| \geq R\) as \(n \to \infty\). Here \(f^n\) denotes the \(n\)-th iterate of \(f\). Let \(I(f)\) be the set of all \(z \in \mathbb{C}\) such that \(|f^n(z)| \to \infty\) as \(n \to \infty\); that is, \(I(f) =\bigcap_{R>0} I_R(f)\). Denote the Hausdorff dimension of a set \( A \subset \mathbb{C}\) by \(\mathrm{HD}(A)\). The authors show that \(\lim_{R\to\infty} \mathrm{HD}(I_R(f)) \leq \frac{2M \rho}{(2 +M \rho)}\). In particular, \(\mathrm{HD}(I(f)) \leq \frac{2M\rho}{(2 +M\rho)}\). Moreover, the authors show, by constructing examples, that these estimates are optimal. 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.