Syllabus: The course aims at presenting billiards as an interesting example of Dynamical systems and, at the same time, to use Billiards as a motivation to develop ideas and techniques that have much larger applicability in the study of the statistical properties of the Dynamical Systems. The prerequisites are minimal knowledge of measure theory, functional analysis, and ergodic theory. However, if asked, I can briefly introduce the missing notions. The program is flexible, depending on students' interests, yet here is a rough outline: Lyapunov exponents and hyperbolicity. How to establish hyperbolicity. Hyperbolic billiards. Gas of hard spheres. Stable and unstable foliations. Ergodicity of piecewise smooth symplectic maps. Quantitative statistical properties for Dynamical Systems and techniques to establish them (e.g., speed of mixing, Central Limit Theorem, Large deviations, ...). How to establish statistical properties of billiards. Billiard flows.
I will try to provide some notes for the material that cannot be easily found in print. However, most of the topics can be found in:
Here are some notes that I hope may help.
I will updated them continuously, without warning, so check for the current version.
Yet, they are most likely full of mistakes, so read at your own risk.
I will try to upload more as we prooceed (but do not keep your hopes too high).