The Lyapunov exponent in the Sinai billiard in the small scatterer limit

Abstract

We show that Lyapunov exponent for the Sinai billiard is $λ= -2\log(R)+C+O(R\log^2 R)$ with $C=1-4\log 2+27/(2π^2)\cdot ζ(3)$ where $R$ is the radius of the circular scatterer. We consider the disk-to-disk-map of the standard configuration where the disks is centered inside a unit square.