Professor of Pure Mathematics

Professor of Pure Mathematics

Ergodic theory and Teichmuller dynamics

My main research interests are in ergodic theory and Teichmuller dynamics, an area of ̈research in dynamical systems which has developed and bloomed in the last decades and that often involves arithmetical, geometrical and combinatorial tools. I am especially interested in parabolic dynamical systems, which are systems which display a "slow" form of chaotic evolution.

Examples of systems studied in Teichmuller dynamics are: ̈

• Polygonal billiards, in which a point-particle moves in a planar polygon bouncing elastically at sides.

• Geodesics on translation surfaces, which are locally Euclidean surfaces but at some conical singularities.

• Maps of the interval which are piecewise isometries, called interval exchange transformations (IETs).

One is interested in investigating ergodic properties that describe how chaotic these systems are.

The properties of these elementary systems are beautifully and deeply connected with the dynamics on an abstract space of deformations, more precisely with the Teichmuller geodesic flow and the SL(2,R) action on moduli spaces. At the level of interval exchange transformations, this connections can be exploited at a more combinatorial level, using a continued fraction algorithm called Rauzy-Veech induction.

Some areas of research for possible PhD projects:

• Area preserving flows on surfaces: in a natural class of flows locally given by a Hamiltonian, I studied properties like mixing and weak mixing; many interesting questions about spectral properties are open;

• Infinite periodic polygonal billiards and periodic infinite translation surfaces; while compact translation surfaces are well understood, the study of infinite covers has just recently started and there are many open questions on the ergodic properties of respectively the billiard and linear flow in this infinite ergodic theory setup.

• Cutting sequences: in a joint work with Smillie, we gave a characterization of the symbolic sequences which code linear trajectories in regular polygons. Similar questions could be addressed in other translation surfaces with the lattice property.

• Interval exchange transformations: questions about distributions of orbit points, as spacings and discrepancy, in the spirit of limit theorems or for special classes of IETs (like bounded type).

If you are interested in knowing more about this research area and potential projects, feel free to contact me by email. You might also want to browse my webpage, in the Slides section you will find slides and videos of talks that I've given explaining my research.