In the last few years, I have worked on problems in ergodic theory, looking at statistical and probabilistic aspects of dynamical systems. This includes systems with no symmetry, compact symmetry and noncompact symmetry. For example, Mike Field, Andrew Török and I show that hyperbolic flows are stably mixing (for open and dense sets of smooth flows). We now have similar results on stable rapid mixing (superpolynomial decay of correlations).
Using renewal theory, I have extended Dolgopyat's ideas on rapid mixing to a class of nonuniformly hyperbolic flows (including the planar periodic Lorentz gas with finite horizons).
Recently, Georg Gottwald and I have developed a new 0-1 test for chaos. This test appears to have many advantages over the standard test of computing the maximal Lyapunov exponent. Click here for a comparison of the 0-1 test and the maximal Lyapunov exponent for the forced van der Pol oscillator. Here is an improved version of the 0-1 test that works well with moderately noisy data (and which seems to compare favourably with the maximal Lyapunov exponent). The mathematical justification for why the test works is provided by the earlier work on ergodic theory mentioned above.
Alistair Windsor and I have an elementary construction of a diffeomorphism with infinitely many attractors with intermingled basins (aka riddled basins).
Previously, my research focused on bifurcation theory/dynamical systems with noncompact symmetry (usually Euclidean symmetry), with particular emphasis on the rigorous foundations of Ginzburg-Landau theory.
As far as compact symmetry groups are concerned, I have worked on
chaotic attractors where the symmetry appears only on average,
robust heteroclinic cycles which provide an elementary mechanism for
producing intermittent phenomena, and on
bifurcation from relative
periodic solutions and discrete
rotating waves (periodic solutions with spatiotemporal symmetry).