The growth rate of an entire function and the Hausdorff dimension of its Julia set

Abstract

Let f be a transcendental entire function in the Eremenko-Lyubich class B. We give a lower bound for the Hausdorff dimension of the Julia set of f that depends on the growth of f . This estimate is best possible and is obtained by proving a more general result concerning the size of the escaping set of a function with a logarithmic tract.1991 Mathematics Subject Classification. 37F10 (primary), 30D05, 30D15 (secondary).

On the packing dimension of the Julia set and the escaping set of an entire function

“…Considerable attention has been paid to the dimensions of Julia sets of entire functions; see [36] for a survey, as well as [3,4,8,9,10,27,28,34] for some recent results not covered there. Many results in this area are concerned with the Eremenko-Lyubich class B consisting of all transcendental entire functions for which the set of critical and finite asymptotic values is bounded.…”

Dynamic rays of bounded-type entire functions

“…This generalizes McMullen's results [McM87] on the escaping sets of exponential and trigonometric functions. Further studies of the Hausdorff dimension of the escaping set for f ∈ B can be found in [BKS09], [RS10].…”

Regularity and growth conditions for fast escaping points of entire functions

“…ε , ε ∈ (0, 1), and R > 0 is such that µ ε (r) > r for r ≥ R. Points that are quite fast escaping arise naturally in complex dynamics and had been used earlier by Bergweiler, Karpińska and Stallard in [5] and by Peter in [12] in results on the Hausdorff measure and Hausdorff dimension of I(f ) and J(f ). It was shown in [17] that there are many classes of functions for which Q(f ) = A(f ).…”