The iterated minimum modulus and conjectures of Baker and Eremenko

TL;DR

This paper introduces a novel approach in transcendental dynamics by studying points whose iterates escape at least as fast as the minimum modulus, leading to new insights on Eremenko's and Baker's conjectures and escape rates.

Contribution

It presents a new method focusing on minimum modulus iterates, providing progress on longstanding conjectures and escape rate analysis in transcendental dynamics.

Findings

Established existence of points escaping to infinity under positive continuous functions.

Provided new results related to Eremenko's conjecture.

Contributed to understanding Baker's conjecture and escape rates in Baker domains.

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 {\it 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.