Remarks on the outer length billiards

TL;DR

This paper investigates outer length billiards, proving periodic versions of the Ivrii conjecture and characterizing invariant curves for tables with periodic points, including explicit constructions for symmetric cases.

Contribution

It establishes 3- and 4-periodic Ivrii conjecture cases and provides a parameterization and construction method for symmetric billiard tables with invariant curves.

Findings

Proved 3- and 4-periodic Ivrii conjecture versions.

Existence of invariant curves for all periods n ≥ 3.

Explicit parameterization and geometric construction for symmetric tables with period 4.

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.