Density of Axiom A in Arnol'd's standard family
Time and place
1 PM on Tuesday, November 24th, 2015;
Lasse Rempe-Gillen (University of Liverpool)
Abstract
In 1961, Arnold introduced a two-dimensional space of self-maps of the circle, as a model of periodically forced nonlinear oscillators exhibiting “phase-locking” phenomena. This family, now known as the standard family, has since served as one of the simplest and most famous models of one-dimensional dynamical systems.
I will discuss a recent result (Duke Math. J. 2015, joint with van Strien) establishing the density of “Axiom A” (or “hyperbolic”) maps in the standard family, solving a long-standing open problem. Here Axiom A systems are those exhibiting the simplest possible type of dynamical behaviour, and the result hence states that – despite the presence of highly “chaotic” examples within the family – regular behaviour is topologically generic. I will also mention connections with recent advances in the dynamics of transcendental entire functions. The talk will be held at a level aimed at a general mathematical audience.
