Billiards with singular invariant curves

TL;DR

This paper constructs smooth convex billiard tables with singular invariant curves, demonstrating that such curves can have controlled singularities and hyperbolic periodic trajectories, advancing understanding of billiard dynamics.

Contribution

It introduces a modified string construction method to produce convex billiard tables with prescribed singular invariant curves and hyperbolic periodic points.

Findings

Singular invariant curves can be precisely controlled and constructed.

Singularities correspond to hyperbolic 2-periodic trajectories.

Invariant curves can have one-sided derivatives at singular points.

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 virtually any closed subset of $\mathbb{S}^1$. 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 $1/2$.