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On the Hausdorff dimension of the escaping set of certain meromorphic functions. (English) Zbl 1294.37018

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.

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37C45 Dimension theory of smooth dynamical systems
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
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