Wednesday 26 Mar 2014AG Dynamics seminar: Density of Axiom A in Arnol'd's standard family

Lasse Rempe-Gillen - University of Liverpool

St Johns room, Kay Building 14:00-15:00

Consider the self-maps of the circle defined by F_{a,b}(t) := t + a + b*sin(2\pi t) (mod 1), where a and b are real parameters, b>0. 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 where the maps are non-invertible. (Axiom A maps are those that 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 gentle recap of one-dimensional real dynamics, including computer simulations, and should be accessible to postgraduate and undergraduate students with an interest in dynamical systems.

[This will be delivered by video-link]

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