Hyperbolicity of asymmetric lemon billiards

Asymmetric lemon billiards was introduced in [CMZZ], where the billiard table $Q(r,b,R)$ is the intersection of two round disks with radii $r\le R$, respectively, and $b$ measures the distance between the two centers. It is conjectured [BZZ] that the asymmetric lemon billiards is hyperbolic when the arc ${\mathrm{\Gamma }}_r$ is a major arc and $R$ is large. In this paper we prove this conjecture for sufficiently large $R$.