On the structure of level sets of uniform and Lipschitz quotient mappings from Rn to R, Geom. Funct. Anal

Abstract. We study two questions posed by Johnson, Lindenstrauss, Preiss, and Schecht-man, concerning the structure of level sets of uniform and Lipschitz quotient mappings from Rn → R. We show that if f: Rn − → R, n ≥ 2, is a uniform quotient mapping then for every t ∈ R, f−1(t) has a bounded number of components, each component of f−1(t) separates Rn 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 R2 to R are locally connected, and we show that for every pair of a constant c> 0 and a function Ω(·) with limr→0 Ω(r) = 0, there exists a natural number M = M(c,Ω), so that for every co-Lipschitz uniformly continuous map f: R2 − → R with a co-Lipschitz constant c and a modulus of uniform continuity Ω, there exists a natural number n(f) ≤ M and a finite set Tf ⊂ R with card(Tf) ≤ n(f)−1 so that for all t ∈ R\Tf, f−1(t) has exactly n(f) components, R2 \ 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 ∈ Tf are also described – they have a finite tree structure. 1