On the Closing Lemma problem for the torus

Simon Lloyd*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


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
Issue number3
Publication statusPublished - Nov 2009
Externally publishedYes


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


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