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

We study the dynamics of area-preserving maps in a non-compact setting. We show that the $C^{\mathrm{\infty }}$-closing lemma holds for area-preserving diffeomorphisms on a closed surface with finitely many points removed. As a corollary, a $C^{\mathrm{\infty }}$-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.