A UNIFORM CONTINUITY PROPERTY OF THE WINDING NUMBER OF SELF-MAPPINGS OF THE CIRCLE

Let u: (formula presented). When u is continuous, it has a winding number deg u, which satisfies (formula presented) and (formula presented). In particular, u → deg u is uniformly continuous for the sup norm. The winding number deg u can be naturally defined, by density, when u is merely VMO. For such u’s, the winding number deg is continuous with respect to the BMO norm. Let (formula presented). In view of the above and of the embedding (formula presented) VMO, maps in (formula presenetd) have a well-defined winding number, continuous with respect to the W1/p,p norm. However, an example due to Brezis and Nirenberg yields sequences (formula presented) such that (formula presented) and (formula presented). Thus deg is not uniformly continuous with respect to the norm (and, a fortiori, with respect to the BMO norm). The above sequences satisfy (formula presented). We prove that a similar phenomenon cannot occur for bounded sequences. More specifically, we prove the following uniform continuity result. Given (formula presenetd) and Μ > 0, there exists some δ = δ(p, Μ) > 0 such that (formula presenetd).