Abstract
Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy–Carleman–Ahlfors theorem implies that if the set of all z for which |f(z)| > R has N components for some R > 0, then the order of f is at least N/2. More precisely, we have log log M(r, f) ≥ (N/2) log r − O(1), where M(r, f) denotes the maximum modulus of f. We show that if f does not grow much faster than this, then the escaping set and the Julia set of f have positive Lebesgue measure. However, as soon as the order of f exceeds N/2, this need not be true. The proof requires a sharpened form of an estimate of Carleman and Tsuji related to the Denjoy–Carleman–Ahlfors theorem.
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M. Aspenberg and W. Bergweiler were supported by the Research Training Network CODY of the European Commission and the G.I.F., the German-Israeli Foundation for Scientific Research and Development, Grant G-809-234.6/2003. W. Bergweiler was also supported by the ESF Research Networking Programme HCAA and the Deutsche Forschungsgemeinschaft, Be 1508/7-1.
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Aspenberg, M., Bergweiler, W. Entire functions with Julia sets of positive measure. Math. Ann. 352, 27–54 (2012). https://doi.org/10.1007/s00208-010-0625-0
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DOI: https://doi.org/10.1007/s00208-010-0625-0