PhD Projects in Ergodic Theory
Research in this area includes probabilistic aspects of deterministic dynamical systems, equilibrium states and thermodynamic formalism, low-dimensional dynamical systems, billiards and Lorentz gases, piecewise isometries, and testing for chaos. Work is also ongoing on the dynamics of discontinuous contracting maps with applications to signal processing, and the application of ergodic theory to systems with symmetry including excitable media. More details, including staff members working in this area, can be found on the Ergodic Theory research group pages.
If you are interested in a project in Ergodic Theory, contact one of the members of the Ergodic Theory research group or the PhD admissions tutor Dr Gianne Derks.
Normal diffusion in compact group extensions of chaotic dynamical
systems (Supervisors: Dr Henk Bruin and Prof Ian Melbourne)
Many classes of dynamical systems (uniformly hyperbolic, nonuniformly hyperbolic, etc) exhibit strong statistical laws such as the central limit theorem and convergence to Brownian motion.
An important class of partially hyperbolic systems occur in systems with symmetry, where the neutral direction along the symmetry group is an added complication.
Previous work (Melbourne & Nicol; Bruin, Holland & Melbourne) indicates that under certain reasonable conditions, strong statistical laws hold for systems with symmetry despite the presence of the neutral direction. This is a topic for further investigation, but perhaps more interesting is the discovery that in some examples the symmetry helps. That is, for certain classes of observables, the system with symmetry might obey the central limit theorem and so on even though the nonsymmetric system does not. (Nonanomalous anomalous diffusion, so to speak.)
Further details: Dr Henk Bruin, Prof Ian Melbourne
Lyapunov exponents for systems with symmetry (Supervisor: Dr Philip Aston, Prof Ian Melbourne)
For uniformly hyperbolic (Axiom A) dynamical systems, it is expected that typically the Lyapunov exponents are distinct. Definitive results are unavailable, but Bonatti & Viana (2004) proved that distinctness of Lyapunov exponents is typical in a related but simpler setting (exponents of linear cocycles).
In systems with symmetry, there are constraints that can force multiple Lyapunov exponents. Results of Aston & Melbourne (2006) on such symmetry-induced multiplicities
are probably optimal, but this has not been pinned down due to the above difficulties of proving distinctness.
This discussion leads naturally to the following problems:
- Extend the multiplicity results of Aston & Melbourne to the setting of linear cocycles (satisfying an appropriate symmetry constraint).
- In the Axiom A case, prove that the results in (i) are sharp using the techniques of Bonatti & Viana.
Further detail: Dr Philip Aston, Prof Ian Melbourne
Testing for chaos, and Wiener sausages (Supervisor: Prof Ian Melbourne)
Gottwald & Melbourne (2004) introduced a new technique for distinguishing chaotic from regular dynamics for deterministic time series data. Let x1, x2, ... ∈ R be a sequence of data from a dynamical system. Choosec>0 and define p(n)=∑nj=1 eijc xj. The claim is that p(n) is bounded if the dynamics is regular and that p(n)∼W(n) if the dynamics is chaotic. Hence it remains to distinguish bounded behaviour from Brownian-motion behaviour.
The method used so far is to use the fact that the growth rate of p(n) is 0 in the regular case and 1 in the chaotic case, but it is important to study other distinguishing features since they may have better convergence properties in practice.
For example, one could study the sausage associated to the process p(n), namely the ε-envelope. There are precise scaling laws for the volume of such sausages, and it would be interesting to (i) establish laws rigorously in the context of the test for chaos, (ii) see how this works for actual time series data.
Statistical properties of nonuniformly hyperbolic systems (Supervisor: Prof Ian Melbourne)
Many classical dynamical systems such as the Lorenz attractor, Hènon-like attractors, dispersing billiards and Lorentz gases, are nonuniformly hyperbolic and modelled by a symbolic system known as a Young tower. Recent work of Melbourne & Terhesiu introduces a new technique for studying the statistical behaviour (central limit theorems with error estimates, stable laws, local limit theorems, large deviations etc) of such systems. The possibilities seem to be endless so it would be a good entrance to a PhD project.
Deterministic random walks in random environments (Supervisor: Prof Ian Melbourne)
The one-dimensional simple symmetric random walk (SSRW) has diffusive √n growth but the introduction of randomness into the environment leads to trapping regions and a much slower subdiffusive (log n)2 growth rate (Sinai, 1972).
These results have a deterministic analogue in systems with a noncompact group of symmetries (the word symmetry here has almost nothing to do with the symmetry in SSRW) where chaotic dynamics in phase space lead to random-like behaviour in the group directions. In particular, √n growth has been proved in this context. In applications (including dynamics of spiral waves in excitable media) the symmetry is not exact and it is reasonable to assume random perturbations in the group directions. In certain situations, it is reasonable to anticipate subdiffusive growth rates.
The project branches from the outset into various distinct directions including:
- Probabilistic: Analyse Sinai diffusion in analogues of the SSRW in random environments with a group of symmetries.
- Deterministic: Analyse Sinai diffusion for deterministic systems in random environments with a group of symmetries.