Mini-courses

• Vincent Delecroix: Triangular billiards

The triangular billiards are among the simplest dynamical systems : a point particle moves without friction inside a triangle and follows the law of reflection of light when hitting a side. Here the dynamics happen in the 3-dimensional unit tangent bundle of the triangle. Despite its simplicity, many questions about triangular billiards remain open. In these three courses I intend to cover the following material (without giving a proof of every result) with an emphasis on current active research and open questions.

Lecture 1. Ergodicity
  a) From billiards to translation surfaces
  b) Ergodicity in rational triangles (following S. Kerckhoff-H. Masur-J. Smillie)
  c) Approximating irrational triangles with rational ones (following Ya. Vorobets)

Lecture 2. Periodic orbits
  a) periodic orbits in irrational triangles (following R. Schwartz)
  b) Siegel-Veech constants

Lecture 3. Mixing
  a) no mixing for translation flows (following A. Katok)
  b) weak-mixing for translation flows (following A. Avila-G. Forni and A. Avila-V. Delecroix)
  c) weak-mixing triangles (following J. Chaika-G. Forni)

• Carlangelo Liverani: Hyperbolic billiards

I will introduce hyperbolic billiards (starting from Sinai and Bunimovich) and then discuss their dynamical properties in increasing order of complexity. For each topic I will try to provide the main ideas and illustrate the technical problems.
Lecture 1: How to establish hyperbolicity
Lecture 2: Ergodicity and mixing (Poincaré map and flow)
Lecture 3: Speed of mixing (standard pairs, functional approach, Hilbert metric)

References:
Chernov, Nikolai; Markarian, Roberto Chaotic billiards. Mathematical Surveys and Monographs, 127. American Mathematical Society, Providence, RI, 2006.
Demers, Mark F.; Zhang, Hong-Kun Spectral analysis of the transfer operator for the Lorentz gas. J. Mod. Dyn. 5 (2011), no. 4, 665–709.
Mark F. Demers, Carlangelo Liverani, Projective Cones for Sequential Dispersing Billiards, preprint     arXiv:2104.06947.

• Olga Paris-Romaskevich: Triangle tiling billiards

I will speak about a class of dynamical systems parametrized by triangles on the plane called triangle tiling billiards and their different aspects : geometric, topological and symbolic. Tiling billiards have been introduced in 2016, and represent a fresh field of mathematics with many open questions. Triangle tiling billiards is a subclass of tiling billiards that has been most studied and I will show different aspects of this mathematical story.

Lecture 1. Triangle tiling billiards and interval exchange transformations with flips (following Baird-Smith, Fromm, Davis and Iyer)
Lecture 2. Triangle tiling billiards and renormalization (following Arnoux-Yoccoz, Rauzy, Hubert&Paris-Romaskevich)
Lecture 3. Triangle tiling billiards and Novikov’s problem (following Dynnikov, Dynnikov&Hubert&Mercat&Skripchenkpo&Paris-Romaskevich)