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arXiv 0811.1089math.DS
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关于环面上有界型向量场的闭引理问题

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

Simon Lloyd

备注：
11 pages, 1 figure
发表信息：
Discrete Contin. Dyn. Syst. 25(3) (2009), 951--962.
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摘要：

研究了二维环面上向量场的开闭引理问题。在有界型旋转数的假设下，证明了光滑向量场的$C^r$闭引理在所有鞍点都是保面积的。即，给定具有非平凡回复点$p$的这种$C^r$向量场$X$，$r\geq 4$，存在任意接近于$C^r$拓扑中的$X$并且通过扭转扰动从$X$获得的向量场$Y$，使得$p$是$Y$的周期点。证明依赖于一个新的结果，在1维动力学的不存在半游荡区间的光滑映射的圆。

英文摘要：

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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