Uniform Quotient Mappings of the PlanebyW

It is shown that if f is a mapping of the plane onto itself that is uniformly continuous with modulus of continuity (r) which is o(p r) as r ! 0 and f is also co-uniformly continuous then f = P h where h is a homeomorphism of the plane and P is a complex polynomial. The same conclusion holds also under other assumptions on the moduli of uniform and co-uniform continuity. However, we also present an example showing that this does not hold for all uniform quotient mappings: There is a mapping of the plane onto itself whose moduli of uniform and co-uniform continuity are both of power type but it maps an interval to zero. We also discuss uniform quotient mappings of the plane onto the line. 1. Introduction Let X and Y be metric spaces. As is well known a mapping f : X ! Y is said to be uniformly continuous if there is a continuous increasing function (r), r 0 with (0) = 0 so that d(f(u); f(v)) (d(u; v)) for all u and v, or in other words, f(B r (x)) B (r) (f(x)) for all x 2 X and r > 0. (B r (x) denotes the open ball with radius r and