Fixed Points of n-Valued Multimaps of the Circle

Robert F. Brown. "Fixed Points of n-Valued Multimaps of the Circle." Bulletin of the Polish Academy of Sciences. Mathematics 54.2 (2006): 153-162. <http://eudml.org/doc/286518>.

@article{RobertF2006,
abstract = {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.},
author = {Robert F. Brown},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {multimap; Nielsen number; fixed point; degree; power map},
language = {eng},
number = {2},
pages = {153-162},
title = {Fixed Points of n-Valued Multimaps of the Circle},
url = {http://eudml.org/doc/286518},
volume = {54},
year = {2006},
}

TY - JOUR
AU - Robert F. Brown
TI - Fixed Points of n-Valued Multimaps of the Circle
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2006
VL - 54
IS - 2
SP - 153
EP - 162
AB - 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.
LA - eng
KW - multimap; Nielsen number; fixed point; degree; power map
UR - http://eudml.org/doc/286518
ER -