Homoclinic and heteroclinic intersections for lemon billiards

We study the dynamical billiards on a symmetric lemon table $\mathcal{Q}(b)$, where $\mathcal{Q}(b)$ is the intersection of two unit disks with center distance $b$. We show that there exists $\delta_0>0$ such that for all $b\in(1.5, 1.5+\delta_0)$ (except possibly a discrete subset), the billiard map $F_b$ on the lemon table $\mathcal{Q}(b)$ admits crossing homoclinic and heteroclinic intersections. In particular, such lemon billiards have positive topological entropy.