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Li, Changbok; Paek, Yujin

Surplus Nielsen type numbers for periodic points on the complement. (English) Zbl 1398.55003

Topology Appl. 244, 31-50 (2018).
Let \(f : (X, A) \to (X, A)\) be a self-map of a pair of compact polyhedra with \(X\) connected. In this paper the authors introduce two new Nielsen type numbers \(SNP_n\left(f;X-A\right)\) and \(SN\phi_n\left(f;X-A\right)\) which provide sharp lower bounds for the minimum number of periodic points of period exactly \(n\) of maps \(g\) that are homotopic to \(f\) as a map of pairs, and that lie in \(X-A\) (\(MP_n\left(f;X-A\right)\)) as well as for the minimum number of periodic points of all periods dividing \(n\) of maps \(g\) that are homotopic to \(f\) as a map of pairs, and that lie in \(X-A\) (\(M\phi_n\left(f;X-A\right)\)) respectively. This is a generalization of surplus fixed point theory by X. Zhao [Topology Appl. 37, No. 3, 257–265 (1990; Zbl 0713.55001)]. The results answer an open question posed by [P. R. Heath and X. Zhao, ibid. 102, No. 3, 253–277 (2000; Zbl 0948.55002)].
Reviewer: João Peres Vieira (Rio Claro)

MSC:

55M20 Fixed points and coincidences in algebraic topology
54H25 Fixed-point and coincidence theorems (topological aspects)

Keywords:

surplus Nielsen class; surplus Nielsen type number; periodic points of maps of pairs; relative Nielsen theory

Citations:

Zbl 0713.55001; Zbl 0948.55002
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References:

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