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Unbounded orbits in C1-generic centrally symmetric strictly convex outer billiards

  • Alfonso Sorrentino
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Abstract

Let ℋ be the space of support functions of centrally symmetric, strictly convex, compact planar bodies whose boundary is a C1 curve, endowed with the C1 topology. We prove that for a residual subset of ℋ the outer billiard maps about the associated bodies admit unbounded orbits escaping to infinity. In particular, this residual set includes support functions arbitrarily C1-close to those of disks or ellipses. This provides an affirmative answer, in the strictly convex C1 setting, to a famous question of Moser and Neumann concerning the existence of unbounded outer-billiard orbits. The key ingredient is a rigidity theorem: if h∈ℋ has a purely singular radius-of-curvature measure, then every continuous invariant tangent graph consists entirely of periodic points. We then show that for a generic table in ℋ these periodic invariant graphs are absent, allowing us to construct escaping orbits via Mather's diffusing mechanism in a Birkhoff region of instability.

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Cite

BibTeXRIS
BibTeX
@misc{sorrentino2026unbounded,
  title = {{Unbounded orbits in $C^1$-generic centrally symmetric strictly convex outer billiards}},
  author = {Alfonso Sorrentino},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06418},
  eprint = {2610.06418},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06418},
}
RIS
TY  - UNPB
TI  - Unbounded orbits in C^1-generic centrally symmetric strictly convex outer billiards
AU  - Alfonso Sorrentino
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06418
UR  - https://arxiv.org/abs/2610.06418
AB  - Let ℋ be the space of support functions of centrally symmetric, strictly convex, compact planar bodies whose boundary is a C^1 curve, endowed with the C^1 topology. We prove that for a residual subset of ℋ the outer billiard maps about the associated bodies admit unbounded orbits escaping to infinity. In particular, this residual set includes support functions arbitrarily C^1-close to those of disks or ellipses. This provides an affirmative answer, in the strictly convex C^1 setting, to a famous question of Moser and Neumann concerning the existence of unbounded outer-billiard orbits. The key ingredient is a rigidity theorem: if h∈ℋ has a purely singular radius-of-curvature measure, then every continuous invariant tangent graph consists entirely of periodic points. We then show that for a generic table in ℋ these periodic invariant graphs are absent, allowing us to construct escaping orbits via Mather's diffusing mechanism in a Birkhoff region of instability.
ER  -