Periodic points on the complement
Nielsen fixed point theory furnishes a lower bound, the Nielsen number \(N(f)\), for the number of fixed points of any map homotopic to a map \(f:X\to X\) of a compact ANR. Nielsen periodic point theory concerns points of period \(n\), that is, fixed points of the \(n\)-th iterate \(f^n:X\to X\) of a map \(f\). The Nielsen number \(N\Phi_n(f)\) is a lower bound for the number of periodic points of all periods dividing \(n\) of maps \(g\) homotopic to \(f\) (which is in general different from \(N(f^n)\) that concerns fixed points of maps \(g\) homotopic to \(f^n\); see [\textit{P. R. Heath}, A survey of Nielsen periodic point theory (fixed \(n)\), in `Nielsen theory and Reidemeister torsion', Proc. Workshop on dynamical zeta functions, Nielsen theory and Reidemeister torsion, Warszawa: Polish Acad. Sci., Inst. Math., Banach Cent. Publ. 49, 159-188 (1999)]. Another Nielsen periodic point number \(NP_n(f)\) does the same with regard to periodic points of minimal period \(n\). The relative Nielsen number \(N(f;X,A)\) [\textit{H. Schirmer}, Pac. J. Math. 122, 459-473 (1986; Zbl 0553.55001)] furnishes a lower bound for the number of fixed points of maps homotopic, as maps of pairs, to a map of a compact ANR pair \(f:(X,A)\to(X,A)\). Relative periodic point numbers for such maps of pairs were developed in [\textit{P. R. Heath}, \textit{H. Schirmer}, and \textit{C. You}, Topology Appl. 63, No. 2, 117-138 (1995; Zbl 0827.55002)]. In the same setting of maps of pairs, there is also a Nielsen fixed point number \(N(f;X-A)\) that furnishes a lower bound for the number of fixed points in \(X-A\) of a map of pairs homotopic as such to \(f\) [\textit{X. Zhao}, Lect. Notes Math. 1411, 189-199 (1989; Zbl 0689.55008)]. The purpose of this paper is to present a theory of Nielsen periodic point numbers of the complement \(N\Phi_n (f;X-A)\) and \(NP_n (f;X-A)\) that give lower bounds for the number of periodic points of periods dividing \(n\), and of minimal period \(n\) respectively, that lie in \(X-A\), for maps of pairs homotopic to \(f:(X,A) \to (X,A)\). When \(X\) is a Jiang space, the paper gives explicit computational results for these Nielsen periodic point numbers in terms of the \(N(f^m;X-A)\) for various values of \(m\). The results in the paper are well illustrated by means of specific examples.
scientific article; zbMATH DE number 1421598