Fixed points of n-valued multimaps of the circle

• Autorzy: Brown R. F.
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

continuous multifunction; Wecken property; degree of n-valued map

PL

multifunkcja ciągła; własność Weckena

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2006
• Tom: Vol. 54, no 2
• Strony: 153--162
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Brown R. F.

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

Bibliografia

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• [3] S. Hu, Homotopy Theory, Academic Press. 1959.
• [4] B. Jiang, On the least number affixed points, Amer. J. Math. 102 (1980), 749-763.
• [5] —, Fixed points and braids, II, Math. Ann. 272 (1985), 249-256.
• [6] B. O'Neill, Induced homology homomorphisms for set-valued maps. Pacific J. Math. 7 (1957), 1179-1184.
• [7] H. Schirmer, Fix-finite approximations of n-valued multif unctions, Fund. Math. 121 (1984), 73-80.
• [8] —, An index and Nielsen number for n-valued multifunctions, ibid. 124 (1984), 207-219.
• [9] H. Schirmer, A minimum theorem for n-valued multifunctions, ibid. 126 (1985), 83-92.
• [10] F. Wecken, Fixpunktklassen, III, Math. Ann. 118 (1942), 544-577.