The iterated minimum modulus and conjectures of Baker and Eremenko

Osborne, John W.; Rippon, Philip J. and Stallard, Gwyneth M. (2019). The iterated minimum modulus and conjectures of Baker and Eremenko. Journal d’Analyse Mathematique, 139(2) pp. 521–558.

DOI: https://doi.org/10.1007/s11854-019-0065-z

Abstract

In transcendental dynamics significant progress has been made by studying points whose iterates escape to infinity at least as fast as iterates of the maximum modulus. Here we take the novel approach of studying points whose iterates escape at least as fast as iterates of the minimum modulus, and obtain new results related to Eremenko's conjecture and Baker's conjecture, and the rate of escape in Baker domains. To do this we prove a result of wider interest concerning the existence of points that escape to infinity under the iteration of a positive continuous function.

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