Fast escaping points of entire functions

Abstract

Abstract. Let f be a transcendental entire function and let A(f ) denote the set of points that escape to infinity 'as fast as possible' under iteration. By writing A(f ) as a countable union of closed sets, called 'levels' of A(f ), we obtain a new understanding of the structure of this set. For example, we show that if U is a Fatou component in A(f ), then ∂U ⊂ A(f ) and this leads to significant new results and considerable improvements to existing results about A(f ). In particular, we study functions for which A(f ), and each of its levels, has the structure of an 'infinite spider's web'. We show that there are many such functions and that they have a number of strong dynamical properties. This new structure provides an unexpected connection between a conjecture of Baker concerning the components of the Fatou set and a conjecture of Eremenko concerning the components of the escaping set.

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

“…In fact, it is shown in [28,Corollary 1.4] that there exists a function f ∈ B for which dim H I(f ) = 1 and in [35] that for every ε > 0 there exists a function f ∈ B such that dim H J(f ) < 1 + ε, and the functions considered in [28,35] satisfy (1.8) as well. On the other hand, for every transcendental entire function f the Hausdorff dimension of I(f ) ∩ J(f ) is at least 1, since this set contains continua; see [29,Theorem 5] and [33,Theorem 1.3].…”

Superattracting fixed points of quasiregular mappings

“…The fast escaping set for transcendental entire functions in the plane has received much recent attention, see for example [15], and the fast escaping set for quasiregular mappings in R n of transcendental type was investigated in [3,5]. For complex polynomials it is clear that I(f ) and A(f ) agree because every point which escapes does so at the same rate.…”

Regularity and growth conditions for fast escaping points of entire functions

“…(Some of these results were shown in [3].) Note that the definition of A(f ) is independent of the choice of R > 0 with the property that M(r) > r for r ≥ R (see [16,Theorem 2…”

“…This set was introduced by Bergweiler and Hinkkanen in [3]. We will use the definition given by Rippon and Stallard in [16] according to which A(f ) = {z : there exists ℓ ∈ N such that |f n+ℓ (z)| ≥ M n (R, f ), for n ∈ N}, where M(r, f ) = M(r) = max |z|=r |f (z)|, for r > 0, and R > 0 is large enough to ensure that M(r) > r for r ≥ R. In the same paper they showed that A(f ) has properties similar to the properties of I(f ) listed above. (Some of these results were shown in [3].)…”