The iterated minimum modulus and conjectures of Baker and Eremenko

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.For Walter Hayman on the occasion of his ninetieth birthday. arXiv:1609.06644v1 [math.CV]

Eremenko’s Conjecture for Functions with Real Zeros: The Role of the Minimum Modulus

“…As mentioned in the introduction, we also showed in [17] that there are large classes of functions for which there exist r > R > 0 such that m n (r) > M n (R) and M n (R) → ∞ as n → ∞.…”

“…We introduced the set V (f ) in [17]. This is the set of points whose iterates grow at least as fast as the iterated minimum modulus; that is, (a) There exists t > 0 such that ϕ n (t) → ∞ as n → ∞.…”

“…In [17] we showed that, if there exist r > R > 0 such that (1.2) m n (r) > M n (R) and M n (R) → ∞ as n → ∞,…”

“…In [17] we asked whether I(f ) is a spider's web whenever the much weaker condition is satisfied that there exists r > 0 such that m n (r) → ∞ as n → ∞. Here we give a positive answer to this question in the case that f is a real function of finite order with only real zeros.…”

“…1. All functions of order less than 1/2 have the property (1.3), along with several other families of functions (see [17,Theorem 1.1]), but many of these functions do not satisfy (1.2). In particular, there are functions of order less than 1/2 that satisfy the hypotheses of Theorem 1.1 but do not satisfy (1.2); indeed, such examples can even be of order 0 (see [22]).…”