Entire functions with Julia sets of positive measure

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 Tsuji related to the Denjoy-Carleman-Ahlfors theorem.

17 pages

References in corpus (4)

• Dynamics of meromorphic functions with direct or logarithmic singularities
• The growth rate of an entire function and the Hausdorff dimension of its Julia set
• On the Hausdorff dimension of the Julia set of a regularly growing entire function
• Hausdorff measure of Julia sets in the exponential family

Cited by in corpus (6)

• Lebesgue measure of escaping sets of entire functions
• Lebesgue measure of escaping sets of entire functions of completely regular growth
• Density of hyperbolicity for classes of real transcendental entire functions and circle maps
• The escaping set in transcendental dynamics
• Lyapunov exponents and related concepts for entire functions
• Speiser meets Misiurewicz