Existence of \(\delta_m\)-periodic points for smooth maps of compact manifold (Q1977493)

Marzantowicz and Przygodski expressed the formula \[ i_m(f)= \sum_{k |m}\mu(k)I (f^{m/k}), \tag{1} \] for the fixed point index \(I(f)\) of \(f\) in terms of periodic points of a compact manifold. If \(i_m(f)\neq 0\), then one says \(m\) is an algebraic period of \(f\). The main goal of this paper is to prove a theorem like (1), but formulated in the language of algebraic periods. The author shows that both theorems are equivalent for the class of maps with finitely many periodic points, but by a use of algebraic periods it is possible to find a \(\delta_m\)-periodic point for maps with infinitely many periodic points as well.