Remarks on the outer length billiards

Abstract

We study outer length billiards; our main results are as follows. We prove 3- and 4-periodic versions of the Ivrii conjecture. We show that, for every period $n\ge 3$, there exists a functional space of billiard tables that possess invariant curves consisting of $n$-periodic points. For $n=4$, we explicitly parameterize such centrally symmetric billiard tables by functions of one variable and describe how to construct these tables geometrically, similarly to the known construction of Radon curves.

Figures (14)

• Figure 1: Outer length billiard rule.
• Figure 2: Computing $l_1, l_2$ in envelope coordinates.
• Figure 3: Invariant symplectic form $dR\wedge d\alpha$.
• Figure 4: $ABC$ is a 3-periodic orbit of the outer length billiard.
• Figure 5: Definition of the distribution $\mathcal{D}_n$.

Theorems & Definitions (21)

• Lemma 2.1
• Lemma 2.2
• Corollary 2.3
• Lemma 2.4
• Corollary 2.5
• Theorem 1
• Remark 2.6
• Lemma 3.1
• Theorem 2
• Lemma 3.2