Lasse Rempe

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

On a question of Eremenko concerning escaping components of entire functions

\mathcal{B}

Topological dynamics of exponential maps on their escaping sets

. 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

Bifurcations in the space of exponential maps

f \in \mathcal{B}

Classification of escaping exponential maps

has no invariant line fields on its escaping set. Finally, we show that any two hyperbolic functions

Siegel disks and periodic rays of entire functions

f,g \in \mathcal{B}

A landing theorem for periodic rays of exponential maps

that belong to the same parameter space are conjugate on their sets of escaping points.

Brushing the hairs of transcendental entire functions

Let f be an entire function with a bounded set of singular values, and suppose furthermore that the postsingular set of f is bounded. We show that every component of the escaping set I(f) is unbounded. This provides a partial answer to a question of Eremenko.

CONNECTED ESCAPING SETS OF EXPONENTIAL MAPS

For the family of exponential maps

Are Devaney hairs fast escaping

E_{\kappa}(z)=\exp(z)+\kappa