Acta Mathematica

Rigidity of escaping dynamics for transcendental entire functions

Lasse Rempe

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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.


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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]
Secondary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX]

2009 © Institut Mittag-Leffler


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.

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