Abstract 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 C^1) 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 S^1 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.
Stefano Baranzini (Sun,) studied this question.