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.
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
- C*-dynamical systems
- Chaos and deterministic behaviour
- Induced dynamical systems
- Jarník and Besicovitch (type) sets
- Low dimensional dynamical systems and their complexity
- Shadowing properties
- Symbolic dynamics
- Tilings and quasicrystals
In addition to this we also work on questions in set-theoretic and general topology, logic and complex systems theory.
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 below
Professor of Mathematics
Chris Good is Professor of Mathematics at the University of Birmingham. He is the author of over 50 research articles in general topology, set-theoretic topology, topological dynamics and mathematical education. He regularly collaborates with mathematicians around the world, including colleagues from Mexico, New Zealand, Poland and the US, as well as from Oxford, and has organised conferences in Birmingham, Oxford and Cambridge.
Tony Samuel is a Lecturer in Mathematics and Statistics at the University of Birmingham. His research interests include aperiodic order, Diophantine approximations, dynamical systems (symbolic and measure preserving), ergodic theory (finite and infinite), geometry (fractal, hyperbolic and non-commutative) and potential theory. Recently, his focus has been on applications of ergodic theory, non-commutative geometry and potential theory to fractals and quasi-crystals. He is also very interested in linking these topics to other areas of mathematics, such as, geometric group theory, geometric measure theory and renewal theory.
Simon Baker is a lecturer in the School of Mathematics. He is interested in topics from Ergodic Theory, Fractal Geometry, and Number Theory. Specific areas include beta expansions, Diophantine approximation, and iterated function systems. He regularly gives invited talks at events held in the UK and abroad.
Sabrina Kombrink is a lecturer of Mathematics. She is interested in geometrically characterising highly irregular objects as well as in the question if one can hear the shape of a fractal drum. Her research lies in the interim of Analysis, Geometry and Stochastics.