A $C^{\infty}$ closing lemma on torus

TL;DR

This paper extends the $C^{ abla}$ closing lemma to area-preserving diffeomorphisms on a torus, demonstrating that for generic such maps, periodic orbits are dense, using flux vectors as a key tool.

Contribution

It introduces a $C^{ abla}$ closing lemma for area-preserving diffeomorphisms on a torus isotopic to identity, generalizing previous Hamiltonian results.

Findings

Density of periodic orbits for generic diffeomorphisms

Use of flux vector as a main tool

Extension of closing lemma to torus case

Abstract

Asaoka & Irie recently proved a $C^{\infty}$ closing lemma of Hamiltonian diffeomorphisms of closed surfaces. We reformulated their techniques into a more general perturbation lemma for area-preserving diffeomorphism and proved a $C^{\infty}$ closing lemma for area-preserving diffeomorphisms on a torus that is isotopic to identity. i.e., we show that the set of periodic orbits is dense for a generic diffeomorphism isotopic to identity area-preserving diffeomorphism on torus. The main tool is the flux vector of area-preserving diffeomorphisms which is, different from Hamiltonian cases, non-zero in general.