Rigidity of escaping dynamics for transcendental entire functions

We prove an analog of Böttcher's theorem for transcendental entire functions in the Eremenko-Lyubich class. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i. e., are quasiconformally equivalent in the sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to the set of points that remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal self-map of the plane. We also prove that the conjugacy is essentially unique. In particular, we show that a function has no invariant line fields on its escaping set. Finally, we show that any two hyperbolic functions f,g that belong to the same parameter space are conjugate on their sets of escaping points.