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Escape rate and Hausdorff measure for entire functions. (English) Zbl 1279.30044

Let \(f\) be an entire function. The escaping set \(I(f)\) of \(f\) is the set of all points which tend to infinity under iteration. Given a sequence \((p_n)\) of positive real numbers tending to infinity, the authors define \(\mathrm{Esc}(f,(p_n))\) to be the set of all \(z\in I(f)\) such that \(|f^n(z)|\leq p_n\) for all large \(n\) and \(\mathrm{Unb}(f,(p_n))=I(F)\setminus\mathrm{Esc}(f,(p_n))\). A gauge function is an increasing, continuous function \(h:[0,\eta)\to [0,\infty)\) satisfying \(h(0)=0\) for some \(\eta>0\). For \(A\subset\mathbb{C}\) and \(\delta>0\) a \(\delta\)-cover of \(A\) is a sequence \((A_j)\) of subsets of \(\mathbb{C}\) with \(\mathrm{diam}(A_j)<\delta\) for all \(j\in\mathbb{N}\) and \(A\subset\bigcup_{j=1}^{\infty}A_j\). Given \(A\subset\mathbb{C}\) and a gauge function \(h\) the Hausdorff measure of \(A\) with respect to \(h\) is given by
\[ H_h(A)=\lim_{\delta\to 0}\inf\left\{\sum_{j=1}^{\infty}h(\mathrm{diam}(A_j)):(A_j)\text{ is a }\delta\text{-cover of }A\right\}. \]
In the paper under review, the authors prove that, if the set of singular values of \(f\) is bounded and \(\lim_{t\to 0}\frac{\log(h(t))}{\log(t)}=1\), the Hausdorff dimension of \(\mathrm{Esc}(f,(p_n))\) with respect to \(h\) is infinite for every sequence \((p_n)\) of positive real numbers tending to infinity.
Furthermore, they show that, if \((p_n)\) is a real sequence tending to infinity and \(h\) a gauge function with \(\lim_{t\to 0}\frac{h(t)}{t}=0\), then there exists an entire function with bounded singular set such that \(H_h(\mathrm{Unb}(f,(p_n)))=0\).
The main methods used are a result by Barański, Karpińska und Zdunik on the hyperbolic dimension of the Julia set of a transcendental meromorphic function with a logarithmic tract and an extension of a result on the Hausdorff dimension of the limit set of iterated function schemes.

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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