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Baker’s conjecture and Eremenko’s conjecture for functions with negative zeros

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Abstract

We introduce a new technique that allows us to make progress on two long standing conjectures in transcendental dynamics: Baker's conjecture that a transcendental entire function of order less than 1/2 has no unbounded Fatou components, and Eremenko's conjecture that all the components of the escaping set of an entire function are unbounded. We show that both conjectures hold for many transcendental entire functions whose zeros all lie on the negative real axis, in particular those of order less than 1/2. Our proofs use a classical distortion theorem based on contraction of the hyperbolic metric, together with new results which show that the images of certain curves must wind many times round the origin.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK

    P. J. Rippon & G. M. Stallard

Authors
  1. P. J. Rippon
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  2. G. M. Stallard
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Correspondence to P. J. Rippon.

Additional information

Both authors were supported by the EPSRC grant EP/H006591/1.

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Rippon, P.J., Stallard, G.M. Baker’s conjecture and Eremenko’s conjecture for functions with negative zeros. JAMA 120, 291–309 (2013). https://doi.org/10.1007/s11854-013-0021-2

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  • DOI: https://doi.org/10.1007/s11854-013-0021-2

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