Generic density of periodic orbits of area-preserving maps on punctured surfaces

TL;DR

This paper proves that generic area-preserving maps on punctured surfaces have dense periodic points and establishes quantitative equidistribution results for such orbits, using advanced tools like PFH Weyl law.

Contribution

It extends the closing lemma to non-compact surfaces and introduces a quantitative equidistribution result for periodic orbits on punctured spheres.

Findings

C^{ abla} -closing lemma holds for punctured surfaces

Generic maps have dense periodic points

Quantitative equidistribution of periodic orbits

Abstract

We study the dynamics of area-preserving maps in a non-compact setting. We show that the $C^{\infty}$-closing lemma holds for area-preserving diffeomorphisms on a closed surface with finitely many points removed. As a corollary, a $C^{\infty}$-generic area-preserving diffeomorphism on such a surface has a dense set of periodic points. For area-preserving maps on a finitely punctured 2-sphere, we establish a more quantitative result regarding the equidistribution of periodic orbits. The proof of this result involves a PFH Weyl law for rational area-preserving homeomorphisms, which may be of independent interest.