Periods and Lefschetz zeta functions
The object of this paper is to prove the following result: If \(f : M \to M\) is a transversal map on a compact manifold whose Lefschetz zeta function \(Z_ f(t)\) has all its poles and zeros equal to roots of unity, and if \(Z_ f(t)\) has an irreducible factor of the form \((1 \pm t^ n)^{\pm 1}\), then (a) \(n\) is a periodic point of \(f\) if \(n\) is odd, and (b) either \(n \over 2\) or \(n\) is a periodic point if \(n\) is even. (A periodic point of \(f\) is a member of the set \(\text{Per}(f) = \{m \in \mathbb{Z}\mid f\) has a periodic orbit of minimal period \(m\)\}.) The assumption that all poles and zeros of the Lefschetz zeta function are roots of unity is satisfied by three basic classes of transversal maps. One is the set of maps whose set of periods \(\text{Per}(f)\) is finite. Another is the self-maps of compact, connected surfaces with \(\text{Per}(f)\) finite or zero topological entropy. The third is the self-maps of the \(n\)-dimensional torus with \(\text{Per}(f)\) finite or zero topological entropy.
scientific article; zbMATH DE number 695063