Generic equidistribution of periodic orbits for area-preserving surface maps

TL;DR

This paper proves that for a generic area-preserving surface diffeomorphism, there exists a sequence of periodic orbits that become uniformly distributed, refining previous density results with a new quantitative approach.

Contribution

It introduces a new quantitative refinement of the generic density theorem for area-preserving surface diffeomorphisms using spectral invariants and variational methods.

Findings

Existence of equidistributed periodic orbits for generic maps

Development of a Weyl law for PFH spectral invariants

Application of variational techniques inspired by minimal hypersurfaces

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.