For the family of exponential maps Eκ(z) = exp(z)+κ, we prove an analog of Böttcher’s theorem by showing that any two exponential maps Eκ1 and Eκ2 are conjugate on suitable subsets of their escaping sets, and this conjugacy is quasiconformal. Furthermore, we prove that any two attracting and parabolic exponential maps are conjugate on their sets of escaping points; in fact, we construct an anal...

We show that the fast escaping set A(f) of a transcendental entire function f has a structure known as a spider’s web whenever the maximum modulus of f grows below a certain rate. We give examples of entire functions for which the fast escaping set is not a spider’s web which show that this growth rate is best possible. By our earlier results, these are the first examples for which the escaping...