Ergodic Theory meeting

Monday 26th March 2018

All talks will be in Room B3.03 in the Zeeman Building (Mathematics Institute).

Schedule

1:30pm-2:30pm Barak Weiss (Tel Aviv)

Title: Random walks on homogeneous spaces and diophantine approximation on fractals

Abstract: We extend results of Y. Benoist and J.-F. Quint concerning random walks on homogeneous spaces of simple Lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily Zariski dense, but satisfies some expansion properties for the adjoint action. Using these dynamical results, we study Diophantine properties of typical points on some self-similar fractals in $\mathbb R^d$. As examples, we show that for any self-similar fractal K satisfying the open set condition (for instance any translate or dilate of Cantor’s middle thirds set or of a Koch snowflake), almost every point with respect to the natural measure on K is not badly approximable. Furthermore, almost every point on the fractal is of generic type, which means (in the one-dimensional case) that its continued fraction expansion contains all finite words with the frequencies predicted by the Gauss measure. Joint work with David Simmons.

2:45pm-3:45pm Selim Ghazouani (Warwick)

Title: Teichmüller dynamics and piecewise affine circle homeomorphisms

Abstract: I will consider families of piecewise affine circle homeomorphisms and ask what dynamical behaviour one should expect from a generic one. One can associate to such piecewise affine maps geometric objects which we call "dilation surfaces". I will try to explain how the geometry of the moduli space of such surfaces relates to the study of the generic piecewise affine circle homeomorphisms. To this end, we will introduce a renormalisation operator called "Teichmüller flow" acting upon the aforementioned moduli space and discuss its dynamical properties.

TEA

4:15pm-5:15pm John Smillie (Warwick)

Title: The horocycle flow in Teichmuller dynamics

Abstract: Teichmuller dynamics can be described as the study of billiard dynamics on rational polygonal tables but it can also be thought of as the struggle to apply homogeneous dynamics tools in a non-homogeneous setting. An interesting current problem is the extent to which Ratner’s theorems about unipotent flows in homogeneous dynamics apply to the horocycle flow in Teichmuller dynamics. I will describe a new example due to Chaika, Weiss and myself which sheds light on this relationship.

For people who arrive in time we will meet in the Common Room around 12:00 to go for lunch in University House (cafeteria style food). We will also go to a restaurant for dinner in the evening.

This is part of the LMS Scheme 3 funded network of collaborative meetings.