## Acta Mathematica

### Rigidity of escaping dynamics for transcendental entire functions

Lasse Rempe

#### Abstract

We prove an analog of Böttcher’s theorem for transcendental entire functions in the Eremenko–Lyubich class $\mathcal{B}$. 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 $f \in \mathcal{B}$ has no invariant line fields on its escaping set. Finally, we show that any two hyperbolic functions $f,g \in \mathcal{B}$ that belong to the same parameter space are conjugate on their sets of escaping points.

#### Note

Supported in part by a postdoctoral fellowship of the German Academic Exchange Service (DAAD) and by EPSRC Advanced Research Fellowship EP/E052851/1.

#### Article information

Source
Acta Math., Volume 203, Number 2 (2009), 235-267.

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.acta/1485892426

Digital Object Identifier
doi:10.1007/s11511-009-0042-y

Mathematical Reviews number (MathSciNet)
MR2570071

Zentralblatt MATH identifier
1226.37027

Rights

#### Citation

Rempe, Lasse. Rigidity of escaping dynamics for transcendental entire functions. Acta Math. 203 (2009), no. 2, 235--267. doi:10.1007/s11511-009-0042-y. https://projecteuclid.org/euclid.acta/1485892426

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