Fatou's example

Abstract

Examples due to P. Fatou and L. Biberbach show that Picard's little theorem cannot be extended to apply to holomorphic mappings of the complex space Cn, n>1. These examples are used to construct a pair of functionally independent entire functions of two complex variables such that the image of C2 by the mapping realized by this pair does not contain any sphere. Here we give an example of the same type which implies a stronger assertion concerning the set of values not taken.