Reflective projective billiards in the plane: odd periods, period five, and metric rigidity

Abstract

We study polygonal projective billiards whose reflection law is constant on each side. We give a single explicit family with an open set of periodic trajectories for every period at least three, including every odd period, and identify the previously known centrally projective examples as a special parameter value. For period five we describe the full scalar-monodromy variety, prove its irreducibility, and show that every configuration preserves a nonzero quadratic form. Under the stated distinct-centre and non-collinearity hypotheses that form is nondegenerate; this yields an explicit unirational classification of the convex five-reflective locus. We also relate the definite members to spherical billiards and prove that, among analytic simply connected constant-curvature surfaces with geodesic walls, only the sphere can carry an open family of periodic trajectories. For general complete analytic ambients with global wall reflections we prove compactness and a common period for the geodesic flow. The arguments reduce the billiard return map to products of harmonic homologies and separate the algebraic closure condition from the strict first-impact inequalities.

Analytic rigidity and symbolic dynamics for two-centre billiards
Stefano Baranzini, Susanna Terracini
arXiv2 Sep 2026PreprintarXiv licenceFull textDynamical systemsarXiv
Polynomial rigidity of strong-field magnetic billiards
Dipesh Bhandari
arXiv19 Oct 2026PreprintCC BY 4.0Full textMathematical physicsarXiv
Irrationality Exponents of Real CM Elliptic Periods and Powers of the Lemniscate Period
Gregory V. Chudnovsky, David V. Chudnovsky
arXiv22 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Rigidity of projective area widths and a weighted Crofton formula on RP³
Tongrui Wang
arXiv23 Sep 2026PreprintarXiv licenceFull textGeometry & topologyarXiv
Period spectrum and rigidity of interval maps
Gabriel Fuhrmann, Maik Gröger, Alejandro Passeggi
arXiv5 Oct 2026PreprintarXiv licenceFull textDynamical systemsarXiv

Cite

BibTeX RIS

BibTeX

@misc{gonzalez2026reflective,
  title = {{Reflective projective billiards in the plane: odd periods, period five, and metric rigidity}},
  author = {Jorge Lucas González},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28817},
  eprint = {2609.28817},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28817},
}

RIS

TY  - UNPB
TI  - Reflective projective billiards in the plane: odd periods, period five, and metric rigidity
AU  - Jorge Lucas González
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28817
UR  - https://arxiv.org/abs/2609.28817
AB  - We study polygonal projective billiards whose reflection law is constant on each side. We give a single explicit family with an open set of periodic trajectories for every period at least three, including every odd period, and identify the previously known centrally projective examples as a special parameter value. For period five we describe the full scalar-monodromy variety, prove its irreducibility, and show that every configuration preserves a nonzero quadratic form. Under the stated distinct-centre and non-collinearity hypotheses that form is nondegenerate; this yields an explicit unirational classification of the convex five-reflective locus. We also relate the definite members to spherical billiards and prove that, among analytic simply connected constant-curvature surfaces with geodesic walls, only the sphere can carry an open family of periodic trajectories. For general complete analytic ambients with global wall reflections we prove compactness and a common period for the geodesic flow. The arguments reduce the billiard return map to products of harmonic homologies and separate the algebraic closure condition from the strict first-impact inequalities.
ER  -

Abstract

Related

Cite