Outer billiards in the complex hyperbolic plane

TL;DR

This work extends the theory of outer billiards to the complex hyperbolic plane $\mathbb{C}H^{2}$ by analyzing a quadratically convex hypersurface $N$ and its exterior $U$. It constructs a double geodesic ray foliation of $U$ via $F(q,r)=\gamma_{J\nu(q)}(r)$, proving that the restrictions to $N\times(0,\infty)$ and $N\times(-\infty,0)$ are diffeomorphisms onto $U$, which yields a well-defined outer billiard map $B$ that is a diffeomorphism and a symplectomorphism with respect to the Kahler form. Special cases show invariance properties under complex lines and geodesic spheres, and the paper provides a framework for studying three-periodic orbits, highlighting open problems in the complex hyperbolic setting. The results unify and extend known properties from the real hyperbolic and complex Euclidean cases to $\mathbb{C}H^{2}$, with potential implications for symplectic dynamics on curved spaces.

Abstract

Given a quadratically convex compact connected oriented hypersurface $N$ of the complex hyperbolic plane, we prove that the characteristic rays of the symplectic form restricted to $N$ determine a double geodesic foliation of the exterior $U$ of $N$. This induces an outer billiard map $B$ on $U$. We prove that $B$ is a diffeomorphism (notice that weaker notions of strict convexity may allow the billiard map to be well-defined and invertible, but not smooth) and moreover, a symplectomorphism. These results generalize known geometric properties of the outer billiard maps in the hyperbolic plane and complex Euclidean space.

Figures (1)

• Figure 1: The outer billiard map in the plane

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Remark 3
• Proposition 4
• Theorem 5
• Lemma 6
• Lemma 7