Integrable Billiards and Related Topics

TL;DR

This survey analyzes integrability across diverse billiard models, framing total integrability as foliations of invariant curves for twist maps and linking it to Hopf rigidity. It synthesizes algebraic, variational, and geometric approaches to the Birkhoff–Poritsky conjecture, highlighting that circles and ellipses emerge as rigid models in multiple settings and that Mather’s $eta$-function provides a powerful inverse-geometric tool. The work unifies results on Birkhoff, outer, symplectic, magnetic, Minkowski, wire, and cone billiards, establishing rigidity, isoperimetric-type inequalities, and effective estimates that quantify proximity to integrable templates. It also outlines a broad agenda of open questions, inviting identification of new integrable examples and deeper understanding of how symmetry, curvature, and norm geometry govern billiard dynamics with practical implications for inverse problems in convex geometry.

Abstract

This paper surveys our results on integrable billiards. We consider various models of billiards, including Birkhoff, outer, magnetic, and Minkowski billiards. Also, we discuss wire billiards and billiards in cones. For four models of convex plane billiards, we also discuss an isoperimetric-type inequality for the Mather $β$-function. We conclude with a section of open questions on this subject.

Figures (9)

• Figure 1: Generating function $L(s_0,s_1)=|\gamma(s_0)-\gamma(s_1)|$
• Figure 2: Invariant curve $\alpha$ of 4-periodic orbits and the region $\mathcal{A}$.
• Figure 3: Outer billiard map
• Figure 4: A symplectic billiard bounce $(x_1,x_2)\mapsto (x_2,x_3)$ in a strictly convex domain $\Omega$.
• Figure 5: Outer length billiard rule, $B_{\Omega4}:M_0\mapsto M_1$.

Theorems & Definitions (29)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Remark 1
• Theorem 5
• Corollary 1
• Theorem 6
• Theorem 7
• Theorem 8