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

Abstract

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.