Fixed Points of $n$-Valued Multimaps of the Circle
A multifunction $\phi \colon X \multimap Y$ is $n$-valued if $\phi(x)$ is an unordered subset of $n$ points of $Y$ for each $x \in X$. The (continuous) $n$-valued multimaps $\phi \colon S^1 \multimap S^1$ 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 $N(\phi)$ of an $n$-valued $\phi \colon S^1 \multimap S^1$ of degree $d$ equals $|n - d|$ and $\phi$ 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 $\phi \colon S^1 \multimap S^1$ of degree $d$ splits into $n$ selfmaps of $S^1$ if and only if $d$ is a multiple of $n$.