%0 Journal Article
%T On the Closing Lemma problem for vector fields of bounded type on the torus
%A Simon Lloyd
%J Mathematics
%D 2008
%I arXiv
%R 10.3934/dcds.2009.25.951
%X We investigate the open Closing Lemma problem for vector fields on the 2-dimensional torus. Under the assumption of bounded type rotation number, the $C^r$ 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 maps of the circle.
%U http://arxiv.org/abs/0811.1089v1