Absence of Absolutely Periodic Euclidean Billiard Orbits

Definitions and mathematical statement

Source conjecture. Euclidean billiards have no absolutely periodic orbits.

The supplied record does not define "absolutely periodic" or specify the dimension, boundary regularity, convexity, or admissible class of billiard tables. Consequently, a fully specified universal conjecture cannot be recovered without further source clarification. The following gives an explicit conditional interpretation consistent with the record's distinction between finite-order degeneracies and infinite-order examples.

In a Euclidean billiard table $Q\subset{\mathbb{R}}^d$, a trajectory travels along straight segments and reflects at a regular boundary collision according to \[ v_{\mathrm{out}}=v_{\mathrm{in}}-2\langle v_{\mathrm{in}},n(x)\rangle n(x), \] where $n(x)$ is a unit normal at the collision point $x$. Let $T$ denote the collision map on collision states consisting of a boundary point and an outgoing unit velocity. Its iterates are considered only where the relevant collisions and return map are defined. For the interpretation below, restrict to non-tangent collisions at smooth boundary points for which the required iterate is $C^\infty$ on a neighborhood of the state.

A periodic state satisfies $T^m(p)=p$ for some integer $m\geq1$; here $m$ need not be the least period. If "absolutely periodic" means that an iterate agrees with the identity to infinite order at a periodic state, its precise local condition is as follows. In local coordinates $\varphi$ near $p$, put $z_0=\varphi(p)$ and \[ F(z)=\varphi\left(T^m(\varphi^{-1}(z))\right)-z. \] The condition is \[ D^\alpha F(z_0)=0 \qquad\text{for every multi-index }\alpha\in{\mathbb{N}}_0^r, \] where $r$ is the dimension of the collision phase space and ${\mathbb{N}}_0=\{0,1,2,\ldots\}$. This includes the periodicity condition when $|\alpha|=0$.

Under this interpretation, the conjecture asserts that for every table in the intended admissible class, every integer $m\geq1$, and every regular periodic state $p$ with a smooth local return map, there exists a multi-index $\alpha$ with $|\alpha|\geq1$ such that \[ D^\alpha F(z_0)\ne0. \] The intended admissible class and the identification of absolute periodicity with this infinite-order condition require confirmation. Vanishing through any prescribed finite order does not by itself establish vanishing at every order.

Short English statement

The source conjectures that Euclidean billiards have no absolutely periodic trajectories. Interpreting this as infinite-order agreement of a local return map with the identity gives the conditional statement above. The intended billiard class and terminology remain unspecified; the supplied partial-resolution assessment is dated curation requiring independent review.

Sources

• Supplied UnsolvedMath local registry record, permanent ID $30004742$, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2021 and names no proposer.
• Supplied report citation: Dynamische Systeme (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/33, pp. 1735--1803. https://doi.org/10.4171/owr/2021/33. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference, labeled "arbitrarily high finite-order orbits": https://arxiv.org/abs/2209.11721.
• Supplied literature reference, labeled "analytic billiard result": https://arxiv.org/abs/math/0110134.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the problem partially solved and says that the no-absolutely-periodic-orbits conjecture remains unresolved; analytic convex cases and periods through four are ruled out, while arbitrarily high finite-order absolutely periodic degeneracies are constructed, rather than infinite-order examples. The supplied background identifies this as literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review. These are supplied editorial assertions, not independent verification of the formulation or mathematics here.