ON THE STRUCTURE OF LEVEL SETS OF UNIFORM AND LIPSCHITZ QUOTIENT MAPPINGS FROM R n TO R

Content Provider	Semantic Scholar
Author	Randrianantoanina, Beata
Copyright Year	2003
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 R n → R. We show that if f : R n −→ 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 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 −→ R. We prove that all level sets of any co-Lipschitz uniformly continuous mapping from R 2 to R are locally connected, and we show that for every pair of a constant c > 0 and a function (� ) with limr!0 (r) = 0, there exists a natural number M = M(c,), so that for every co-Lipschitz uniformly continuous map f : R 2 −→ R with a co-Lipschitz constant c and a modulus of uniform continuity , 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, R 2 \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 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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Alternate Webpage(s)	http://arxiv.org/pdf/math/0301367v1.pdf
Alternate Webpage(s)	http://www.users.miamioh.edu/randrib/levelsets.pdf
Alternate Webpage(s)	http://www.users.miamioh.edu/randrib/levelsetsshort.pdf
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article