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arXiv math/0301367math.FAmath.GN
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从${\mathbb{R}}^n$到${\mathbb{R}}$的一致Lipschitz商映射的水平集结构

On the structure of level sets of uniform and Lipschitz quotient mappings from ${\mathbb{R}}^n$ to ${\mathbb{R}}$

Beata Randrianantoanina

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34 pages, 10 figures
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摘要：

我们研究了Johnson，Lindenstrauss，Preiss和Schechtman提出的两个问题，关于$R^n\to R$中一致和Lipschitz商映射水平集的结构。证明了如果$f:R^n\to R$，$n\geq 2$是一个一致商映射，则对每个$t\in R$，$f^{-1}(t)$有一个有界的分支数，$f^{-1}(t)$的每个分支分离$R^n$，并且分支数的上界仅依赖于$n$和$f$的协一致和一致连续模.其次，我们得到了任意遗传局部连通、局部紧、无端点、无简单闭曲线的连通集的形式的一个刻画，并应用它描述了co-Lipschitz一致连续映射$f:R^2\to R$的水平集的结构.本文证明了从$R^2$到$R$的任何co-Lipschitz一致连续映射的所有水平集都是局部连通的，并且证明了对于每一对常数$c>0$和具有$\lim_{r\to 0}Ω(r)=0$的函数$Ω$，存在一个自然数$M=M(c,Ω)$，使得对于每一个具有co-Lipschitz常数$c$和一致连续模$Ω$的co-Lipschitz一致连续映射$f:R^2\to R$，存在自然数$n(f)\le M$和具有$\card(T_f)\leq n(f)-1$的有限集$T_f\subset R$，使得对于所有$t\in R\setminus T_f$，$f^{-1}(t)$恰好具有$n(f)$分量，$R^2\setminus f^{-1}(t)$恰好具有$n(f)+1$分量，并且$f^{-1}(t)$的每个分量与实线同胚，并且将平面分成恰好2个分量。还描述了$f^{-1}(s)$的$f^{-1}(s)$组件的数量和形式-它们具有有限的图形结构。我们给出了一个例子，从$R^2\to R$具有非局部连通水平集的均匀商映射。

英文摘要：

We study two questions posed by Johnson, Lindenstrauss, Preiss, and Schechtman, concerning the structure of level sets of uniform and Lipschitz quotient maps from $R^n\to R$. We show that if $f:R^n\to R$, $n\geq 2$, is a uniform quotient map then for every $t\in R$, $f^{-1}(t)$ has a bounded number of components, each component of $f^{-1}(t)$ separates $R^n$ and the upper bound of the number of components depends only on $n$ and the moduli of co-uniform and uniform continuity of $f$. Next we obtain a characterization of the form of any closed, hereditarily locally connected, locally compact, connected set with no end points and containing no simple closed curve, and we apply it to describe the structure of level sets of co-Lipschitz uniformly continuous mappings $f:R^2\to R$. We prove that all level sets of any co-Lipschitz uniformly continuous map from $R^2$ to $R$ are locally connected, and we show that for every pair of a constant $c>0$ and a function $Ω$ with $\lim_{r\to 0}Ω(r)=0$, there exists a natural number $M=M(c,Ω)$, so that for every co-Lipschitz uniformly continuous map $f:R^2\to R$ with a co-Lipschitz constant $c$ and a modulus of uniform continuity $Ω$, there exists a natural number $n(f)\le M$ and a finite set $T_f\subset R$ with $\card(T_f)\leq n(f)-1$ so that for all $t\in R\setminus T_f$, $f^{-1}(t)$ has exactly $n(f)$ components, $R^2\setminus f^{-1}(t)$ has exactly $n(f)+1$ components and each component of $f^{-1}(t)$ is homeomorphic with the real line and separates the plane into exactly 2 components. The number and form of components of $f^{-1}(s)$ for $s\in T_f$ are also described - they have a finite graph structure. We give an example of a uniform quotient map from $R^2\to R$ which has non-locally connected level sets.

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