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Fixed Points of n-Valued Multimaps of the Circle

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Brown R.

Bulletin of the Polish Academy of Sciences. Mathematics

2006

tom 54

nr 2

153-162

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.

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1 Bulletin of the Polish Academy of Sciences. Mathematics

1 Brown R.

1 2006

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Fixed Points of n-Valued Multimaps of the Circle