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: S¹ ⊸ S¹ 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: S¹ ⊸ S¹ 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: S¹ ⊸ S¹ of degree d splits into n selfmaps of S¹ if and only if d is a multiple of n.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba54-2-7,
author = {Robert F. Brown},
title = {Fixed Points of n-Valued Multimaps of the Circle},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {54},
year = {2006},
pages = {153-162},
zbl = {1108.55003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba54-2-7}
}
Robert F. Brown. Fixed Points of n-Valued Multimaps of the Circle. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 54 (2006) pp. 153-162. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba54-2-7/