Generic equidistribution for area-preserving diffeomorphisms of compact surfaces with boundary

TL;DR

This paper proves that generic area-preserving diffeomorphisms on compact surfaces with boundary have equidistributed periodic orbits and dense periodic points, extending mean action inequalities to all such surfaces.

Contribution

It establishes generic equidistribution and density of periodic points for area-preserving diffeomorphisms on surfaces with boundary, extending previous results to broader classes.

Findings

Generic diffeomorphisms have equidistributed periodic orbits

Periodic points are dense for generic area-preserving maps

Extension of mean action inequalities to all compact surfaces with boundary

Abstract

We prove that a generic area-preserving diffeomorphism of a compact surface with non-empty boundary has an equidistributed set of periodic orbits. This implies that such a diffeomorphism has a dense set of periodic points, although we also give a self-contained proof of this "generic density'' theorem. One application of our results is the extension of mean action inequalities proved by Hutchings and Weiler for the disk and annulus to generic Hamiltonian diffeomorphisms of any compact surface with boundary.