Generic Equidistribution of Periodic Orbits for Area-Preserving Surface Maps

Abstract

Abstract We prove that a $C^{\infty }$-generic area-preserving diffeomorphism of a closed, oriented surface admits a sequence of equidistributed periodic orbits. This is a quantitative refinement of the recently established generic density theorem for area-preserving surface diffeomorphisms. The proof has two ingredients. The first is a “Weyl law” for PFH spectral invariants, which was used to prove the generic density theorem. The second is a variational argument inspired by the work of Marques–Neves–Song and Irie on equidistribution results for minimal hypersurfaces and three-dimensional Reeb flows, respectively.