On the Closing Lemma problem for vector fields of bounded type on the torus

Details On the Closing Lemma problem for vector fields of bounded type on the torus On the Closing Lemma problem for vector fields of bounded type on the torus

2008-11-07

arxiv.org/abs/0811.1089v1

Discrete Contin. Dyn. Syst. 25(3) (2009), 951--962.

Mathematics

Dynamical Systems

11 pages, 1 figure

Scientific paper

10.3934/dcds.2009.25.951

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.

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