Fixed Points of $n$-Valued Multimaps of the Circle

Robert F. Brown¹

¹ Department of Mathematics University of California Los Angeles, CA 90095-1555, U.S.A.

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 54 (2006) no. 2, pp. 153-162

Résumé

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 $n$.

Détail

DOI : 10.4064/ba54-2-7

Keywords: multifunction phi colon multimap n valued phi unordered subset points each continuous n valued multimaps phi colon multimap classified homotopy integer valued degree nielsen fixed point theory multimaps due schirmer nielsen number phi n valued phi colon multimap degree equals phi homotopic n valued power map has exactly fixed points wecken property schirmer established manifolds dimension least three holds circle n valued multimap phi colon multimap degree splits selfmaps only multiple nbsp