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

Huadi Qu; Zhihong Xia

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arXiv · 2106.08844

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

Huadi Qu·Zhihong Xia

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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.

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BibTeX RIS

Huadi Qu, Zhihong Xia. 2021-06-16. A $C^{\infty}$ closing lemma on torus. https://arxiv.org/abs/2106.08844

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