Pure Mathematics Seminar: Lasse Rempe-Gillen

Lasse Rempe-Gillen, Liverpool University

Wednesday 29 January 2014, 1600-1700
A54, Postgraduate Statistics Centre Lecture Theatre

Density of Axiom A in Arnol'd's standard family

Consider the self-maps

12(t):ℝ/ℤ → ℝ/ℤ , t → t+μ12sin(2πt)

of the circle. (Here μ1 ∈ ℝ and μ2 >0 are parameters.) This family, known as the standard family, was introduced by Arnol'd in 1961 to model periodically forced nonlinear oscillators, and has since served as one of the simplest models of one-dimensional dynamical systems. I will discuss a recent result (joint with van Strien) establishing the density of Axiom A (or hyperbolic) maps in the region {μ2 >1/(2π)}, where the maps are non-invertible. (Axiom A maps exhibit the simplest type of dynamical behaviour.) This solves a long-standing open problem. I will also mention connections with recent advances in the dynamics of transcendental entire functions. The talk will begin with a short general introduction to one-dimensional dynamics (and the standard family) for a general mathematical audience.