Richard Sharp
Title: Reconstructing functions from periodic orbit data
Abstract: Hyperbolic flows are chaotic flows which, in particular, have a countable infinity of periodic orbits forming a dense subset of the phase space. Given a sufficiently regular real-valued function, a natural question is to what extent one can reconstruct it if one knows its integrals around all periodic orbits. This was completely answered by Livsic in the early 1970s and introduced the concept of dynamical cohomology. More recently, in the context of Anosov flows (a class which includes geodesic flows over compact negatively curved manifolds), A. Gogolev and F. Rodriguez Hertz, studied the situation where one knows the integrals only for null-homologous periodic orbits, showing that then Livsic's result is modified by the addition of an `"abelian" term coming from a closed 1-form. I obtained short proof of this result, based on thermodynamic formalism, and showed that it still holds if one restricts to periodic orbits with a trivial lift with respect to a regular amenable covering. Subsequently, M. Pollicott and I showed that a more classical approach gives the same conclusion only requiring the lifted flow on the cover to be topologically transitive.
Giulio Tiozzo
Title:
Singular harmonic measures for random walks on cocompact Fuchsian and Kleinian groups
Abstract:
The question of the singularity at infinity of the hitting measure of random walks has a long history, originating from the work of Furstenberg in the 1960s. In 2011, Kaimanovich and Le Prince conjectured that the hitting measure of any finitely supported random walk on a discrete subgroup Γof SLN(ℝ) is singular at infinity with respect to the Lebesgue measure. Using algebraic and geometric convergence and hyperbolic Dehn filling, we prove the singularity conjecture for certain measures on cocompact Fuchsian and Kleinian groups. Joint work with N. Bogachev and P. Kosenko.
Casper Oelen
Title: Non-Birkhoff periodic orbits in symmetric planar billiards.
Abstract: A mathematical billiard is a dynamical system describing the free motion of a point mass inside a domain, subject to elastic reflections at the boundary. Such systems are simple to describe yet exhibit a wide range of dynamical behaviours.
A classical result of Poincaré and Birkhoff implies that every smooth, strictly convex planar billiard admits at least two periodic orbits of any rational rotation number. These are called Birkhoff orbits, and they admit a simple geometric description – they are homeomorphic to regular (star) polygons. In this talk, we discuss the existence of non-Birkhoff periodic orbits in symmetric convex planar billiards – orbits that lack the ordering property characteristic of regular (star) polygons. We present a quantitative criterion for the existence of such orbits with prescribed period, rotation number, and spatio-temporal symmetry. Furthermore, we show that arbitrarily small analytic perturbations of the circular billiard admit non-Birkhoff periodic orbits of any rational rotation number. This is joint work with Bob Rink (VU Amsterdam) and Mattia Sensi (University of Trento).
