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In transcendental dynamics significant progress has been made by studyingpoints whose iterates escape to infinity at least as fast as iterates of themaximum modulus. Here we take the novel approach of studying points whoseiterates escape at least as fast as iterates of the {\it minimum} modulus, andobtain new results related to Eremenko's conjecture and Baker's conjecture, andthe rate of escape in Baker domains. To do this we prove a result of widerinterest concerning the existence of points that escape to infinity under theiteration of a positive continuous function.

The iterated minimum modulus and conjectures of Baker and Eremenko