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Hausdorff measure of the escaping parameter without endpoints is zero for exponential family. (English) Zbl 1410.37047

The Julia set of \(\lambda e^z\), where \(0<\lambda <1/e\), is a so-called Cantor bouquet, consisting of uncountably many pairwise disjoint curves (called Devaney hairs or rays) which connect a point in the plane (called the endpoint) with \(\infty\).
C. McMullen [Trans. Am. Math. Soc. 300, 329–342 (1987; Zbl 0618.30027)] showed that the Julia set has Hausdorff dimension \(2\), and B. Karpińska [C. R. Acad. Sci., Paris, Sér. I, Math. 328, No. 11, 1039–1044 (1999; Zbl 0955.37024)] proved the surprising result that the Hausdorff dimension of the set of hairs without their endpoints is \(1\). The latter result was sharpened by W. Bergweiler and J. Wang [Math. Z. 281, No. 3–4, 931–947 (2015; Zbl 1343.37024)] who showed that the Hausdorff measure of this set with respect to the gauge function \(h(t)=t/(\log(1/t))^s\) is equal to 0 if \(s>1\).
In this paper a corresponding result for the parameter space is obtained. Using the parametrization \(E_\kappa(z)=e^z+\kappa\) the authors consider the set \(J=\big\{\kappa:E_\kappa^n (\kappa)\to \infty \ \text{as}\ n\to\infty\big\}\). M. Förster and D. Schleicher [Ergodic Theory Dyn. Syst. 29, No. 2, 515–544 (2009; Zbl 1160.37364)] proved that \(J\) consists of hairs and W. Qiu [Acta Math. Sin., New Ser. 10, No. 4, 362–368 (1994; Zbl 0817.30011)] proved the analogue of McMullen’s result, i.e., that \(J\) has Hausdorff dimension 2. M. Bailesteanu et al. [Nonlinearity 21, No. 1, 113–120 (2008; Zbl 1142.37033)] proved the analogue of Karpińska’s result, namely that if \(E\) denotes the set of endpoints of the hairs in \(J\), then \(J\backslash E\) has Hausdorff dimension 1.
The main result of the current paper is that \(J\backslash E\) has Hausdorff measure \(0\) with respect to the gauge function \(h(t)=t/(\log(1/t))^s\) if \(s>1\).

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
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References:

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