Ergodic theory of group extensions and geometric rigidity. Many physical systems have some form of symmetry which often determines the behaviour of trajectories of the system. The ergodic theory of systems with compact Lie group symmetry (in particular compact group extensions of hyperbolic systems) has undergone a resurgence in the last ten or so years as part of a programme to understand partially hyperbolic systems. The picture for systems with noncompact Lie group symmetry is not as clear, and efforts have concentrated on the classification of noncompact group extensions via geometric rigidity theorems, in particular Livsic theorems. Mark Pollicott and I have studied aspects of geometric rigidity in noncompact group extensions of ergodic systems. In particular we are interested in the question of when the existence of a measurable solution to a cohomological equation posed on a group extension of an ergodic dynamical system implies the existence of a continuous solution. This question is crucial not only to the classification of the group extension but also to its mixing and stability properties. As part of a research programme to understand the dynamics of systems with noncompact symmetry, Peter Ashwin, Ian Melbourne and I have studied the dynamical properties of a class of noncompact group extensions --- in particular Euclidean group extensions --- by considering such extensions as abelian extensions of compact group extensions. This programme has had success in explaining the behaviour of spiral wave solutions to partial differential equations which model reactive media and certain chemical reactions. We are particularly interested in the statistical properties of such sytems, such as the existence of central limit theorems and Brownian motion like behaviour.
Ergodic theory of hyperbolic dynamical systems. Hyperbolic dynamical systems are widely accepted as the standard model for physical systems exhibiting chaotic behaviour. Under mild assumptions hyperbolic dynamical systems possess the Bernoulli property and in a mathematical sense they are equivalent to a system formed by tossing a coin or spinning a roulette wheel. I have explored methods to establish the Bernoulli property in chaotic systems and also investigated the stability properties of Bernoulli dynamical systems under random perturbation. The orbit structure (but not individual orbits) of these most random systems displays a high degree of stability to quite general random perturbations. The question of stability is fundamental in any mathematical model and an understanding of a systems response to random or deterministic perturbation is necessary if the model is to be fully understood or useful. The stability of chaotic systems also has an important biological and physical role. An important notion in dynamical systems theory is that of a chaotic attractor- a stable set which governs the gross statistics (such as time averages of observables) of a large set of initial conditions in a dynamical system yet has exponential divergence of nearby individual orbits. Many biological systems, such as a beating heart, display chaotic behaviour yet the collection of orbits as a whole is stable. In recent joint work with Ian Melbourne I have investigated the statistical properties of endomorphisms, using transfer operator techniques.
Dynamics of skew products, iterated function systems and other topics. Many of the mathematical models I have studied have a skew product structure. In collaborations with David Broomhead, Demetris Hadjiloucas and Charles Walkden I have studied the regularity and stability of invariant graphs in skew products with appropriately defined negative Lyapunov exponents. This work has applications to the study of Iterated Function Systems (IFS), which are used in computer graphics and data compression. In joint work with Markus Adahl and Ian Melbourne I have studied the statistical behaviour of random compositions of isometries acting on $\R^n$.
My other research interests have been in the dynamics associated with invariant subspaces, in particular mechanisms inducing on-off intermittent behaviour. Invariant subspaces or submanifolds often play an organisational role for the dynamics of the entire system. The main goal here is to analyse models which give rise to intermittency to produce a quantitative understanding of how the ergodic properties (the density of invariant measures, mixing properties, distribution of orbits, distribution of laminar phases) depend upon the parameters of the system. This work is of interest to engineers and physicists since it is basically the problem of understanding how and when systems stay synchronised. Understanding the synchronisation of coupled oscillators has been a major impetus to this line of research.
Equivariant dynamical systems. During postdoctoral work at the University of Houston and the University of Warwick I became interested in equivariant dynamical systems (or symmetric dynamical systems) and studied the topological and statisical properties of attractors in systems with symmetry. If a map or flow has, for example, a reflectional symmetry then this has strong implications for the mixing properties of the system and the possible symmetries of any attractors the map or flow may have. The goal in this area is to learn as much as possible about the behaviour of the system and the properties of the dynamics that it exhibits just by knowing the symmetries of the system.