Combinatorial scheme of finding minimal number of periodic points for smooth self-maps of simply connected manifolds (Q360533)

Let \(M\) be a closed smooth connected and simply connected manifold with \(m:=\dim M\geq4\) and \(f:M\to M\) a \(C^1\)-map. The authors are interested in the minimal number of \(r\)-periodic points in the \(C^1\)-homotopy class of \(f\), i.e. the minimal number \(D_r^m[f]\) of fixed points of \(g^r\) where \(g\) is \(C^1\)-homotopic to \(f\). Invoking an earlier result obtained by the authors [Topology Appl. 158, No. 3, 276--290 (2011; Zbl 1211.55004)]) one may find \(g\) such that in addition all \(r\)-periodic points of \(g\) are fixed points and \(\#\text{Fix}(g^r)= D_r^m[f]\). Define \(\text{reg}_k(n)=k\) if \(k|n\) and \(=0\) else. A result by the second author and \textit{W. Marzantowicz} [Homotopy methods in topological fixed and periodic points theory. Topological Fixed Point Theory and Its Applications 3. Berlin: Springer (2006; Zbl 1085.55001)] states that for a fixed point \(x_0\) of \(f\) we have that \(\text{ind}(f^n,x_0)=\sum_{k=1}^\infty a_k\text{reg}_k(n)\) where \(a_n=\frac1n\sum_{k|n}\mu(k)\text{ind}(f^{(n/k)},x_0)\). The authors then write the Lefschetz number as \(L(f^n)=\sum_{k|n}b_k\text{reg}_k(n)\) and assume that \(r\) is odd and \(b_k\not=1\) whenever \(k\not=1\) divides \(r\). In this situation they present a scheme of calculating \(D_r^m[f]\). In case \(r\) is a product of different odd primes they provide explicit formulae for \(D_r^m [f]\) and provide lower and upper bounds for this quantity.