The statistics of the trajectory in a certain billiard in a flat two-torus

Abstract

We consider a billiard in the punctured torus obtained by removing a small disk from the two-dimensional flat torus, with trajectory starting from the center of the puncture. In this case the phase space is given by the range of the velocity only. We prove that the probability measures on $[0,\infty )$ associated with the first exit time are weakly convergent when the size of the puncture tends to zero and explicitly compute the density of the limit.

21 pages, 6 figures

References in corpus (2)

• The Lyapunov exponent in the Sinai billiard in the small scatterer limit
• The average length of a trajectory in a certain billiard in a flat two-torus

Cited by in corpus (10)

• On the distribution of free-path lengths for the periodic Lorentz gas III
• Kinetic transport in the two-dimensional periodic Lorentz gas
• Free path lengths in quasicrystals
• The average length of a trajectory in a certain billiard in a flat two-torus
• Power-law distributions for the free path length in Lorentz gases
• Free path lengths in quasi crystals
• On the Farey fractions with denominators in arithmetic progression
• Distribution of the linear flow length in a honeycomb in the small-scatterer limit
• The distribution of rationals in residue classes
• A density theorem on even Farey fractions