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Rigidity of escaping dynamics for transcendental entire functions

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Acta Mathematica

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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, U.K.

    Lasse Rempe

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  1. Lasse Rempe
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Correspondence to Lasse Rempe.

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

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Rempe, L. Rigidity of escaping dynamics for transcendental entire functions. Acta Math 203, 235–267 (2009). https://doi.org/10.1007/s11511-009-0042-y

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  • DOI: https://doi.org/10.1007/s11511-009-0042-y

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