Dynamical Systems Seminar

We consider a Sinai billiard table in the \(d\) dimensional unit torus with a single spherical obstacle of radius \(< 1/2\). Extending the billiard table periodically to the entire plane, we obtain a periodic Lorentz gas of unbounded free flight. In case \(d=2\), elementary geometry shows that a long flight of length \(T\) is typically followed by a flight of length \(T^{1/2}\). In this talk, we extend this result to arbitrary dimension \(d\). Our main result is that a long flight of length \(T\) is typically followed by a flight of length \(T^{1/d}\) in any dimension \(d\). Furthermore, we describe the limiting stochastic process obtained by rescaling the first flight time by \(T\), the second one by \(T^{1/d}\), etc. Our approach is based on the theory of Marklof and Strombergsson which uses homogeneous dynamics. This is a joint work with Xingyu Liu.