Properties of Lipschitz quotient mappings on the plane
Cristina Villanueva Segovia · 2018
In the present work, we are concerned with the relation between the Lipschitz and co-Lipschitz constants of a mapping f : ℝ2 → ℝ2 and the cardinality of the inverse image of a point under the mapping f, depending on the norm on ℝ2. It is known that there is a of real numbers 0 < ... < Pn <…< P1 < 1 such that for any Lipschitz quotient mapping from the plane to itself, if the ratio between the co-Lipschitz and Lipschitz constants of f is bigger than Pn, then the cardinality of any fibre of f is less than or equal to n. Furthermore, it is proven that for the case the values of this are Pn = 1/n + 1) for each n ∈ ℕ and that these are sharp. A natural question is: given a normed space (ℝ2 , II · II) whether it is possible to find the values of the 0 pn II · II implies #f- 1 (x) ≤ n for all x ∈ ℝ2. In this work we prove that the same Euclidean scale, Pn = 1/(n+1), works for any norm on the plane. Here we follow the general idea in a previous paper by Maleva but verify details carefully. On the other hand, the question whether this is sharp leads to different conclusions. We show that for some non-Euclidean norms the Euclidean scale is not sharp, but there are also non-Euclidean norms for which a Lipschitz quotient exists satisfying max# f - 1(x) = 2 and c/L = 1/2.