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** with superpolynomial decay
of correlations
for open and dense sets of C^{r} flows, r at least 2.

Using operator renewal theory, I have extended Dolgopyat's ideas
on rapid mixing to a class of **nonuniformly hyperbolic flows** (including
suspended Hénon attractors). An additional
truncation step yields results for
**slowly mixing flows** including optimal results for the
infinite horizon planar periodic Lorentz gas with circular obstacles.
Subsequent results with Péter Bálint deal with the general
infinite horizon planar periodic Lorentz gas. (Still being written up.)
Bálint and I also proved superpolynomial decay
of correlations for the flow in the case of billiards with cusps.

Dalia Terhesiu and I have extended the theory of operator renewal sequences to
the ** infinite ergodic theory** setting, and have obtained results
on convergence of the transfer operator, and second order asymptotics, for
a large class of systems with infinite measure, including intermittency maps
and parabolic rational maps of the complex plane.

Also with Terhesiu, as a byproduct of an approach that failed in the infinite case (so far), we have developed a simplified treatment of decay of correlations and statistical limit laws for systems modelled by Young towers in the finite measure case. We recover existing results for subexponential tails and obtain improved estimates for irregularly varying tails.

Matthew Nicol and I have proved the **almost sure invariance principle**
for nonuniformly hyperbolic flows for both
scalar
and
vector
valued observables. For example, particles in
the finite horizon planar periodic Lorentz gas
undergo approximately two-dimensional Brownian motion.
Mark Holland and I
prove analogous results for the Lorenz
attractor.

Roland Zweimüller and I have proved results on the weak invariance principle
in the presence of a stable law, applying to systems such as Pomeau-Manneville
intermittency maps. Convergence fails in the standard
J_{1} Skorokhod topology but holds in the
M_{1} Skorokhod topology.

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).
Further significant improvements have led to the (final?)
implementation.
The mathematical justification for why the test works is provided by the earlier
work on ergodic theory mentioned above.

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.

Click here for a survey article on rigorous results concerning structure and universality of reduced equations in spatially-extended systems.

As far as compact symmetry groups are concerned, I have worked on
symmetries of
chaotic attractors where the symmetry appears only on average,
on
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).

Click here and here for work of David Chan (Ph. D. student) on bifurcations from rotating waves in systems with spherical symmetry. Some important results for systems with E(3) symmetry remain to be written up.

Preprints

Meetings 2012 onwards