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 mappings from $\mathbb{R}^n \rightarrow \mathbb{R}$. We show that if $f:\mathbb{R}^n \rightarrow \mathbb{R}, n\geq 2$, is a uniform quotient mapping then for every $t \in \mathbb{R}, f^{-1}(t)$ has a bounded number of components, each component of $f^{-1}(t)$ separates $\mathbb{R}^n$ 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 prove that all level sets of any co-Lipschitz uniformly continuous mapping from $\mathbb{R}^n$ to $\mathbb{R}$ are locally connected, and we show that for every pair of a constant $c > 0$ and a function $\Omega(\cdot)$ with $\lim_{{r\to 0}} \Omega (r)=0$, there exists a natural number $M = M (c, \Omega)$, so that for every co-Lipschitz uniformly continuous map $f:\mathbb{R}^n \rightarrow \mathbb{R}$ with a co-Lipschitz constant $c$ and a modulus of uniform continuity $\Omega$, there exists a natural number $n(f) \leq M$ and a finite set $T_f \subset \mathbb{R}$ with card$(T_f)\leq n(f)-1$ so that for all $t \in \mathbb{R} \setminus T_f, f^{-1}(t)$ has exactly $n(f)$ components, $\mathbb{R}^2 \setminus f^{-1}(t)$ 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 tree structure.