Billiards With Singular Invariant Curves

We investigate the regularity of invariant curves of rotation number $1/2$ for a special class of symplectic twist maps of the annulus, billiard maps. We construct strictly convex smooth tables close to the circle having singular (i.e., not C¹) invariant curves. Our method relies on a modification of the classical string construction and allows precise control over the location of singularities: they form a discrete set whose closure can contain any closed subset of S¹ with empty interior. Each singularity corresponds to a hyperbolic $2$-periodic trajectory and the invariant curves admit distinct one-sided derivatives at these points. An analogous construction yields perturbations of constant-width tables with invariant curves of rotation number $1/2$.