Mathematische Annalen

, Volume 352, Issue 1, pp 27–54

# Entire functions with Julia sets of positive measure

Article

## 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 rO(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.

### Mathematics Subject Classification (2000)

Primary 37F10 Secondary 30D05 30D15

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## Copyright information

© Springer-Verlag 2010

## Authors and Affiliations

1. 1.Mathematisches Seminar, Christian-Albrechts-Universität zu KielKielGermany