A counterexample to Eremenko's conjecture (Lasse Rempe, University of Manchester)

17.02.2026 10:30

Let f be an entire function (i.e., a holomorphic self-map of the complex plane), and suppose that f is transcendental, i.e., not a polynomial. The *escaping set* of f consists of those points that tend to infinity under repeated application of f. (For example, all real numbers belong to the escaping set of the exponential map, since they tend to infinity under repeated exponentiation.) In 1989, Eremenko conjectured that every connected component of the escaping set is unbounded.

Eremenko's conjecture has been a central problem in transcendental dynamics. Prior to our most recent work, a number of stronger versions of the conjecture had been disproved, while weaker ones had been established, and the conjecture has also been shown to hold for a number of classes of functions. I will describe recent work with David Martí-Pete and James Waterman in which we construct a counterexample to the conjecture. The talk should be accessible to a general mathematical audience, including postgraduate students.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-05, Tuesday 17.02.2026, Séminaire "Groupes et géométrie"

entrée libre