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KAM splittings and equidistributed periodic orbits for stable hypersurfaces

  • Robert Cardona
  • Oliver Edtmair
  • Rohil Prasad
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Abstract

We show that any stable hypersurface of a symplectic $4$-manifold, on which the cohomology class of the symplectic form restricts to a multiple of a rational class, can be C∞-approximated by (possibly unstable) hypersurfaces whose closed characteristics equidistribute. The cohomological condition is necessary due to a famous example of Herman. The proof combines KAM theory with recent quantitative closing lemmas for Reeb flows and area-preserving maps. As a further application, we prove that every geodesible volume-preserving vector field on a closed three-manifold can be C∞-approximated by volume-preserving vector fields with equidistributed periodic orbits.

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Cite

BibTeXRIS
BibTeX
@misc{cardona2026splittings,
  title = {{KAM splittings and equidistributed periodic orbits for stable hypersurfaces}},
  author = {Robert Cardona and Oliver Edtmair and Rohil Prasad},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25465},
  eprint = {2609.25465},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25465},
}
RIS
TY  - UNPB
TI  - KAM splittings and equidistributed periodic orbits for stable hypersurfaces
AU  - Robert Cardona
AU  - Oliver Edtmair
AU  - Rohil Prasad
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.25465
UR  - https://arxiv.org/abs/2609.25465
AB  - We show that any stable hypersurface of a symplectic $4$-manifold, on which the cohomology class of the symplectic form restricts to a multiple of a rational class, can be C^∞-approximated by (possibly unstable) hypersurfaces whose closed characteristics equidistribute. The cohomological condition is necessary due to a famous example of Herman. The proof combines KAM theory with recent quantitative closing lemmas for Reeb flows and area-preserving maps. As a further application, we prove that every geodesible volume-preserving vector field on a closed three-manifold can be C^∞-approximated by volume-preserving vector fields with equidistributed periodic orbits.
ER  -