About us
The mathematical theory of dynamical systems is a vital part of modern mathematical analysis, where new theoretical developments have been inspired by applications in physics and nature. Thus, this theory forms a strong bridge between pure and applied mathematics as well as many other areas of the natural sciences.
Our research focuses mainly on studying topological and ergodic aspects of dynamical systems and their interactions with problems occurring within algebra, combinatorics, geometry, number theory and physics.
The fundamental research theme is to study (continuous) functions mapping (compact) topological spaces to themselves. As the function is iterated (repeatedly applied), asymptotic structures appear for example periodic orbits, recurrent and non-wandering points, as well as minimal sets.
Results from ergodic theory can be used to develop a understanding of such structures, points and sets, by taking a probabilistic viewpoint to investigate the average statistical behaviour of the system. Here, the central objects of study are measures, this leads to the question which ones are important, and what happens for typical points for these measures under repeated application of given map? This naturally leads to applications and developments of multifractals and the thermodynamic formalism as pioneered by David Ruelle and Rufus Bowen. Taking things one step further, one can then begin to study the speed at which the system begins to look completely random, a key signature of what is nowadays known as chaotic behaviour.
We also study non-Hausdorff topological spaces that arise in order theory, most notably, continuous domains. Dana Scott introduced domains as a mathematical model of computation. Domain theory has its roots in topological algebra, and has enriched computer science with powerful methods from topology, order theory, and category theory. Our main aim is to develop domain-theoretic frameworks for some of the most important problems of classical and modern mathematics that arise in functional analysis, ordinary and partial differential equations, probabilistic programming, and machine learning, to name but a few.
Areas of active research within our group include, but are not limited to, the following.
- The Auslander-Yorke dichotomy
- β-transformations and interval maps
- Bifurcation theory
- Chaos and deterministic behaviour
- Domain theory
- Fractal geometry
- Induced dynamical systems
- Jarník and Besicovitch (type) sets
- Low dimensional dynamical systems and their complexity
- Shadowing properties
- Stone duality
- Symbolic dynamics
- Tilings and quasicrystals
- Thermodynamic formalism
We welcome enquiries from potential research students, postdocs and visitors who wish to work within our group. Specific research interests of the members of the group are given on our staff page.