On Lipschitz ball noncollapsing functions and uniform co-Lipschitz mappings of the plane (Q2495021)

A map \(f\) between two metric spaces is co-Lipschitz, with constant \(C\), if the image of a ball with centre \(x\) and radius \(r\) always contains the ball with centre \(f(x)\) and radius \(Cr\). Reversing this inclusion obviously yields the definition of a Lipschitz map. If we assume only that the image of a ball of radius \(r\) always contains a ball with radius \(Cr\), then \(f\) is called ball noncollapsing. The author had previously shown [Geometric aspects of functional analysis. Proceedings of the Israel seminar (GAFA) 2001--2002. Berlin: Springer. Lect. Notes Math. 1807, 148-157 (2003; Zbl 1058.46053)] that if \(f\) is an \(L\)-Lipschitz and \(C\)-co-Lipschitz map from the plane to itself, then for every point \(x\), the cardinality of \(f^{-1}(x)\) is at most \(L/C\). Here she reaches the same conclusion, under the weaker assumption that \(f\) is uniformly continuous and \(C\)-co-Lipschitz. This time \(L\) is replaced by the weak Lipschitz constant of \(f\), which exists because uniformly continuous maps between normed spaces are Lipschitz for large distances. The \(n\)th power function on the complex plane shows that this estimate is sharp. Another interesting result is that any Lipschitz and co-Lipschitz map from the plane to itself does not collapse areas of measurable sets. In the previous paper, it had also been shown that if \(f\) is an \(L\)-Lipschitz, \(C\)-ball noncollapsing map of the line (or the plane) to itself with \(L<2C\), then \(f\) is one-one; the absolute value function on the line shows that this estimate is sharp. In a real tour de force it is proved here that there is an \(L\)-Lipschitz, \(C\)-ball noncollapsing map \(f\) of the line to itself and a point \(x\) such that \(f^{-1}(x)\) is infinite if, and only if, \(L>3C\).