Statistical Properties of Hyperbolic Groups and Dynamics

The project is at the interface of ergodic theory, a branch of mathematical analysis that studies dynamical systems and, more generally, group actions, from a probabilistic point of view, and the area of geometry that studies moduli spaces, i.e. spaces that parametrise geometric structures. An example of the latter is the so-called Teichmueller space that parametrises the hyperbolic metrics on a compact surface of fixed genus or, equivalently, representations of the fundamental group into a matrix group. The aim of the project is to further our understanding of the statistical properties of the discrete groups that arise as points in moduli spaces and more general groups of similar type -- a field which has grown up over the last few years, though building on the well developed ergodic theory of the chaotic dynamical systems, geodesic flows, which are associated to the geometry. For example, Calegari and Fujiwara showed that certain natural classes of observable satisfy a Central Limit Theorem, i.e. that they behave as if they were derived from a simple random system such as coin tossing. For the special case of compact surfaces, where the moduli space is the Teichmueller space, Pollicott and Sharp obtained more precise probabilistic results such as the Almost Sure Invariance Principle, which deals with the approximation of random variables by Brownian motion, and a local version of the Central Limit Theorem. An objective of the research is to extend these more precise results to a wider class of groups. The methodology would be based on an analysis of the spectral properties of families of so-called transfer operators that are part of the thermodynamic formalism associated to symbolic dynamical systems that encode information about the group actions. To date (September 2018), the student has (a) completed work on a central limit theorem for periodic orbits of hyperbolic (chaotic) flows and (b) succeeded in extending the results of Calegari-Fujiwara to a wider class of observables for general non-elementary hyperbolic groups . The work in (b) is close to being completed and the students has obtained both central limit results and large deviations results. There is scope for the student to pursue further investigation of more refined results such as local limit theorems or statistical invariance principles. The potential beneficiaries are researchers in ergodic theory, geometric group theory, hyperbolic geometry and chaotic dynamical systems. The research is in the research areas of Geometry and Topology and Mathematical Analysis, and is wholly within the Mathematical Sciences theme. .