Least number of \(n\)-periodic points of self-maps of \(PSU(2)\times PSU(2)\)
Let \(f : M \to M\) be a self-map of a compact manifold and \(n \in \mathbb{N}.\) \textit{B. Jiang} [Lectures on Nielsen fixed point theory. Providence, RI: American Mathematical Society (AMS) (1983; Zbl 0512.55003)] introduced a number, \(NF_n(f),\) called Nielsen-Jiang periodic number of \(f.\) Also, G. Graff and J. Jezierski introduced a number, \(NJD_n(f),\) called Nielsen-Jiang-Dold number of \(f\), [\textit{G. Graff} and \textit{J. Jezierski}, Topology Appl. 158, No. 3, 276--290 (2011; Zbl 1211.55004)]. If \(dim(M) \geq 3\) then these numbers satisfy \(NF_n(f) = \min\{\#Fix(g^{n})\mid g \sim f, \,\, g \,\, continuous \}\) and \(NJD_n(f) = min\{\#Fix(g^{n})\mid g \sim f, \,\, g \,\, smooth \}.\) In general \(NF_n(f) < NJD_n(f).\) The main question is to know when \(NF_n(f) = NJD_n(f)\)? If \(f\) is essentially reducible, has the Jiang property and \(L(f^{k}) = det(I - A^{k})\) for an integer matrix \(A,\) then a necessary condition for the equality \(NF_n(f) = NJD_n(f)\) can be shown. In this paper the author considers \(M = PSU(2),\) the projective unitary group, or \(M = PSU(2) \times PSU(2),\) and shows that this necessary condition is also sufficient. The proof of this fact is made using the description of the graph of orbits of Reidemeister classes \(\mathcal{G} \mathcal{O} \mathcal{R}(f^{n}).\)
scientific article; zbMATH DE number 7468365
scientific article