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arXiv · 2609.25465

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 $C^\infty$-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 $C^\infty$-approximated by volume-preserving vector fields with equidistributed periodic orbits.

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Robert Cardona, Oliver Edtmair, Rohil Prasad. 2026-09-21. KAM splittings and equidistributed periodic orbits for stable hypersurfaces. https://arxiv.org/abs/2609.25465

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