Julia and Escaping Set Spiders’ Webs of Positive Area

Abstract

Abstract. We study the dynamics of a collection of families of transcendental entire functions which generalises the well-known exponential and cosine families. We show that for functions in many of these families the Julia set, the escaping set and the fast escaping set are all spiders' webs of positive area. This result is unusual in that most of these functions lie outside the EremenkoLyubich class B. This is also the first result on the area of a spider's web.

Lebesgue measure of escaping sets of entire functions of completely regular growth

“…One consequence of this result is that Sixsmith's result [31] concerning functions of the form (1.1) or (1.2) remains valid if the a k occurring there are polynomials instead of constants. Indeed, such functions satisfy a linear differential equation with constant coefficients.…”

Lebesgue measure of Julia sets and escaping sets of certain entire functions

“…McMullen's result on the measure of J(sin(αz + β)) has been extended to various classes of functions in [2,7,34]. In this paper we give another criterion for the Julia set or escaping set of an entire function to have positive measure.…”

Exponential polynomials with Fatou and non-escaping sets of finite Lebesgue measure

“…In fact, it can be seen from the proof that J (f ) ∩ A(f ) also has positive measure. Sixsmith [14] proved that if f (z) = q j =1 a j exp(ω j q z), where q ≥ 2, a j ∈ C\{0}, and ω q = exp(2πi/q), then J (f ) ∩ A(f ) has positive measure. Sixsmith remarked without proof that his result remains true for f (z) = q j =1 a j exp(b j z), ( 1 ) where q ≥ 3, a j , b j ∈ C\{0}, arg(b j ) < arg(b j +1 ) < arg(b j ) + π for j ∈ {1, .…”