Baker's conjecture for functions with real zeros

Abstract

Baker's conjecture states that a transcendental entire function of order less than 1/2 has no unbounded Fatou components. It is known that, for such functions, there are no unbounded periodic Fatou components and so it remains to show that they can also have no unbounded wandering domains. Here we introduce completely new techniques to show that the conjecture holds in the case that the transcendental entire function is real with only real zeros, and we prove the much stronger result that such a function has no orbits consisting of unbounded wandering domains whenever the order is less than 1. This raises the question as to whether such wandering domains can exist for any transcendental entire function with order less than 1. Key ingredients of our proofs are new results in classical complex analysis with wider applications. These new results concern: the winding properties of the images of certain curves proved using extremal length arguments, growth estimates for entire functions, and the distribution of the zeros of entire functions of order less than 1.

The iterated minimum modulus and conjectures of Baker and Eremenko

“…We prove part (b) of Theorem 1.3 by showing that if the conditions in part (a) hold, then the set A R (f ), defined earlier, is a spider's web, a property that has several strong consequences [31]. This is, in essence, the approach used to prove all partial results on Baker's conjecture prior to the recent papers [21,32]; see [2,16,17,30] and the discussion in [32,Introduction].…”

Iterating the Minimum Modulus: Functions of Order Half, Minimal Type

“…Finally, we remark that consideration of the iterates of the minimum modulus first arose in connection with the conjecture of Baker that entire functions of order 1/2, minimal type, have no unbounded Fatou components; see [7] for the most recent progress on this conjecture.…”

Patrick Denis Barry (1934-2021)

“…such that m(r, f ) is greater than M (r, f ) raised to a fixed positive power for at least one value of r in each interval of the form [R, R σ ] for R sufficiently large. The latter result has made Paddy's cos πρ -type theorem a favourite tool in complex dynamics for studying a conjecture of Noel Baker that an entire function f of order less than 1/2 cannot have unbounded components of its Fatou set, the open set where the iterates of f form a normal family; see [27] for a survey on the history of Baker's conjecture and [31] for more recent developments.…”