On the Closing Lemma problem for the torus

Simon Lloyd*

*Corresponding author for this work

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Abstract

We investigate the open Closing Lemma problem for vector fields on the 2-dimensional torus. The local Cr Closing Lemma is verified for smooth vector fields that are area-preserving at all saddle points. Namely, given such a Cr vector field X, r ≥ 4, with a non-trivially recurrent point p, there exists a vector field Y arbitrarily near to X in the Cr topology and obtained from X by a twist perturbation, such that p is a periodic point of Y. The proof relies on a new result in 1-dimensional dynamics on the nonexistence of semi-wandering intervals of smooth order-preserving circle maps.

Original languageEnglish
Pages (from-to)951-962
Number of pages12
JournalDiscrete and Continuous Dynamical Systems
Volume25
Issue number3
DOIs
Publication statusPublished - Nov 2009
Externally publishedYes

Keywords

  • Black cell
  • Cherry flow
  • Denjoy theorem
  • Wandering interval

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Lloyd, S. (2009). On the Closing Lemma problem for the torus. Discrete and Continuous Dynamical Systems, 25(3), 951-962. https://doi.org/10.3934/dcds.2009.25.951