Properties of Lipschitz quotient mappings on the plane

• Main Author: Villanueva Segovia, Cristina
• Published: University of Birmingham 2018
• Subjects: 510; QA Mathematics
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.753026

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 scale of real numbers 0 < ... < Pn <...< P1 < 1 such that for any Lipschitz quotient mapping from the Euclidean 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 Euclidean case the values of this scale 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 scale 0 < . . . < pn II · II < ... < p1 II · II < 1 such that for any Lipschitz quotient mapping from (R2, II · II) to itself, with Lipschitz and co-Lipschitz constants equal to L and c respectively, the relation c/L > 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 scale 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.