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2009 Rigidity of escaping dynamics for transcendental entire functions
Lasse Rempe
Author Affiliations +
Acta Math. 203(2): 235-267 (2009). DOI: 10.1007/s11511-009-0042-y


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.

Funding Statement

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


Download Citation

Lasse Rempe. "Rigidity of escaping dynamics for transcendental entire functions." Acta Math. 203 (2) 235 - 267, 2009.


Received: 11 March 2008; Published: 2009
First available in Project Euclid: 31 January 2017

zbMATH: 1226.37027
MathSciNet: MR2570071
Digital Object Identifier: 10.1007/s11511-009-0042-y

Primary: 37F10
Secondary: 30D05

Rights: 2009 © Institut Mittag-Leffler

Vol.203 • No. 2 • 2009
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