FIXED POINTS OF n-VALUED MULTIMAPS OF THE CIRCLE

A multifunction φ : X ( Y is n-valued if φ(x) is an unordered subset of n points of Y for each x ∈ X. The (continuous) n-valued multimaps φ : S ( S are classified up to homotopy by an integer-valued degree. In the Nielsen fixed point theory of such multimaps, due to Schirmer, the Nielsen number N(φ) of an n-valued φ : S ( S of degree d equals |n − d| and φ is homotopic to an n-valued power map that has exactly |n−d| fixed points. Thus the Wecken property, that Schirmer established for manifolds of dimension at least three, also holds for the circle. An n-valued multimap φ : S ( S of degree d splits into n selfmaps of S if and only if d is a multiple of n. Subject Classificaton 55M20; 54C20, 55M25