DEPARTMENT of  MATHEMATICS   Pure Research Ergodic Theory and Dynamical Systems

Welcome to our Homepage!

Research Interests  |  Seminars  |  Staff  |  Other Links  |

The Ergodic Theory and Dynamical Systems group at Manchester is very active. We run a regular departmental seminar, together with a series of collaborative meetings with neighbouring institutions. There are many opportunities for postgraduate study within the group.

Dynamical systems has, at its core, the study of the global orbit structure of maps and flows. Thus we want to take a point in space, and understand what happens to it as it evolves in time. In general, this is impossible ('sensitive dependence on initial conditions') - an observation that has given rise to the vast new area of applied dynamics (also called nonlinear science, perhaps more popularly known as chaos theory).

However, dynamical systems is also closely intertwined with most major areas of pure mathematics: topology, differential geometry, hyperbolic geometry, functional analysis, number theory, geometric group theory, commutative algebra, to name but a few.

Ergodic theory is the study of dynamics in the presence of an invariant measure. Most dynamical systems have a very complicated orbit structure; ergodic theory allows us to describe the long-term behaviour of a 'typical' orbit. Ergodic theory is very closely related to dynamical systems and has also been used to study many problems in other areas of mathematics.

Our Research Interests

At Manchester, our research interests span a wide range of topics: hyperbolic and partially hyperbolic systems, connections with geometric and algebraic structures, negatively curved manifolds, geodesic flows, zeta functions, surface homeomorphisms and rotation sets, symbolic dynamics and thermodynamic formalism, geometric group theory and Julia sets.

Here are some of the things that we have been working on recently:

The construction and study of ergodic properties of invariant measures of geodesic flows and horocycle flows and flame flows on general symmetric spaces.  Applications to linear actions of 2 x 2 matrices over a wide variety of fields.

Estimates on the analytic and meromorphic domains of zeta functions and Poincare series associated to negatively curved manifolds and hyperbolic flows.

Using transfer operator techniques to effectively estimate the Hausdorff dimension of certain dynamically defined sets.  Applications to Julia sets of quadratic polynomials and to limit sets of Kleinian groups.

The study of the distribution of periodic orbits of hyperbolic systems, particularly with respect to homology.  Applications to closed geodesics and automorphic forms.

Statistical properties of dynamical systems: estimating rates of mixing, approximation by Brownian motion.

Studying the structure and ergodic properties of group extensions of Anosov and Axiom A dynamical systems, using techniques from the theory of partially hyperbolic dynamical systems and thermodynamic formalism.

Seminars

There is an active dynamical systems seminar at Manchester.  In addition, there is a regular series of collaborative meetings between the dynamical systems groups of Liverpool University, Manchester University, Queen Mary, University of London and Surrey University. These meetings are funded by a Scheme 3 grant from the London Mathematical Society. For details of the most recent meeting, click here.

Members of Staff involved:

Mark Holland (RA)

Richard Oudkerk (RA)

Mark Pollicott

Richard Sharp

Charles Walkden

Alistair Windsor (RA)

Links

Click here for some dynamical systems links!