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September  2009, 25(3): 951-962. doi: 10.3934/dcds.2009.25.951

On the Closing Lemma problem for the torus

Simon Lloyd 1,

1. 

School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

Received  December 2008 Revised  April 2009 Published  August 2009

We investigate the open Closing Lemma problem for vector fields on the $2$-dimensional torus. The local $C^r$ Closing Lemma is verified for smooth vector fields that are area-preserving at all saddle points. Namely, given such a $C^r$ vector field $X$, $r\geq 4$, with a non-trivially recurrent point $p$, there exists a vector field $Y$ arbitrarily near to $X$ in the $C^r$ 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 non-existence of semi-wandering intervals of smooth order-preserving circle maps.
Keywords: Cherry flow, Black cell., Wandering interval, Denjoy theorem.
Mathematics Subject Classification: Primary: 34D30; Secondary: 37E35, 37C1.
Citation: Simon Lloyd. On the Closing Lemma problem for the torus. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 951-962. doi: 10.3934/dcds.2009.25.951
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