Fatou’s web

Abstract

Let f be Fatou's function, that is, f (z) = z + 1 + e −z. We prove that the escaping set of f has the structure of a`spider's web' and we show that this result implies that the non-escaping endpoints of the Julia set of f together with innity form a totally disconnected set. We also present a well-known transcendental entire function, due to Bergweiler, for which the escaping set is a spider's web and we point out that the same property holds for some families of functions.

Eremenko’s Conjecture for Functions with Real Zeros: The Role of the Minimum Modulus

“…The next result we need concerns uniform rates of escape of quite a general form. This result was proved in [6,Theorem 1.4]. Note that, although the statement of [6, Theorem 1.4] assumes that the sequence (a n ) satisfies a n+1 ≤ M(a n ), for n ∈ N, the proof there only uses the consequence of this assumption that a n ≤ M n (R) for n ∈ N and some R > 0, and we now state the result in that form.…”

Escaping Endpoints Explode

“…Evdoridou [15] has recently announced a result that implies that the set of non-escaping points of the Fatou function z → z + 1 + e −z (whose Julia set is a Cantor bouquet), together with ∞, is totally separated. The proof can be adopted to show also that the set of non-escaping endpoints-and even the set of all endpoints not belonging to the fast escaping set A( f )-of an exponential map with Cantor bouquet Julia set has the same property.…”

On the connectivity of the escaping set in the punctured plane

“…There exist several examples of transcendental entire functions with a connected escaping set; for example, this is the case for the exponential function [Rem11]. Furthermore, for many functions I(f ) is a spider's web, that is, a connected set that separates every point of C from ∞; see, for example, [Evd16]. Rippon and Stallard [RS11] also proved that I(f ) ∪ {∞} is a connected subset of C; note that this does not rule out the possibility that I(f ) has a bounded component.…”

“…The function f (z) = sin z is an example for which J(f ) is connected [Dom97] (but not a spider's web [Osb13b]) and I(f ) is disconnected as R ⊆ C \ I(f ). On the other hand, for Fatou's function f (z) = z + 1 + e −z we know that J(f ) is disconnected (it is an uncountable union of disjoint curves), but I(f ) is connected (in fact, it is a spider's web [Evd16]).…”