arXiv 1404.13022014-04-04math.DS

映射刚性扰动下不动点的持久性

Persistence of fixed points under rigid perturbations of maps

Salvador Addas-Zanata, Pedro A. S. Salomão

摘要：

设$f:S^1\times [0,1]\to S^1\times [0,1]$是一个与恒等映射同伦且保持面积形式的实解析环同态。假设对于某个提升$\tilde {f}:\mathbb{R}\times [0,1]\rightarrow \mathbb{R}\times [0,1]$，我们有${\rm Fix}(\tilde{f})=\mathbb{R}\times \{0\}$，并且$\tilde{f}$正向平移$\mathbb{R}\times \{1\}$中的点。令$\tilde{f}_ε$为刚性水平平移$(x,y)\mapsto (x+ε,y)$对$\tilde{f}$的扰动。我们表明，对于所有的$ε>0$足够小，我们有${\rm Fix} (\tilde{f}_ε)=\emptyset $。证明如下从Kerékjártó的建设布劳威尔线的方向保持同胚的平面没有不动点。这个结果在正则性假设方面是尖锐的：存在一个满足上述所有性质的自同构$f$，除了$f$不是实解析的，而只是光滑的，所以上面的结论是假的。这种映射是通过生成函数构造的。

英文摘要：

Let $f:S^1\times [0,1]\to S^1\times [0,1]$ be a real-analytic annulus diffeomorphism which is homotopic to the identity map and preserves an area form. Assume that for some lift $\tilde {f}:\mathbb{R}\times [0,1]\rightarrow \mathbb{R}\times [0,1]$ we have ${\rm Fix}(\tilde{f})=\mathbb{R}\times \{0\}$ and that $\tilde{f}$ positively translates points in $\mathbb{R}\times \{1\}$. Let $\tilde{f}_ε$ be the perturbation of $\tilde{f}$ by the rigid horizontal translation $(x,y)\mapsto (x+ε,y)$. We show that for all $ε>0$ sufficiently small we have ${\rm Fix} (\tilde{f}_ε)=\emptyset $. The proof follows from Kerékjártó's construction of Brouwer lines for orientation preserving homeomorphisms of the plane with no fixed points. This result turns out to be sharp with respect to the regularity assumption: there exists a diffeomorphism $f$ satisfying all the properties above, except that $f$ is not real-analytic but only smooth, so that the above conclusion is false. Such a map is constructed via generating functions.

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摘要：

英文摘要：

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