Topological dynamics of exponential maps on their escaping sets

Abstract

For the family of exponential maps $E_{\kappa}(z)=\exp(z)+\kappa$, we prove an analog of Böttcher's theorem by showing that any two exponential maps $E_{\kappa_1}$ and $E_{\kappa_2}$ 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 analog of Douady's ‘pinched disk model’ for the Julia sets of these maps. On the other hand, we show that two exponential maps are generally not conjugate on their sets of escaping points. We also answer several questions about escaping endpoints of dynamic rays. In particular, we give a necessary and sufficient condition for the ray to be continuously differentiable in such a point, and show that escaping points can escape arbitrarily slowly. Furthermore, we show that the principle of topological renormalization is false for attracting exponential maps.