Periodic points on nilmanifolds and solvmanifolds
Let \(M\) be a compact manifold and \(f:M \to M\) a self map on \(M\). For any natural number \(n\), the \(n\)-th iterate of \(f\) is the \(n\)-fold composition \(f^ n:M \to M\). The fixed point set of \(f\) is \(\text{fix} (f)=\{x \in M:f(x)=x\}\). We say that \(x \in M\) is a periodic point of \(f\) is \(x\) is a fixed point of some \(f^ n\) and we denote the set of all periodic points of \(f\) by \(\text{per} (f)=\bigcup^ \infty_{n=1} \text{fix} (f^ n)\). We say that \(\text{per} (f)\) is homotopically finite, denoted \(\text{per} (f) \sim\) finite, iff there is a \(g\) homotopic to \(f\) such that \(\text{per} (g)\) is finite. When \(M\) is a torus B. Halpern has shown that \(\text{per} (f) \sim\) finite iff the sequence of Nielsen numbers \(\{N(f^ n)\}^ \infty_{n=1}\) is bounded. The main objective of this work is to extend these results to all nilmanifolds and to consider to what extent they can be extended for compact solvmanifolds. A compact nilmanifold is a coset space of the form \(M=G/ \Gamma\) where \(G\) is a connected simply connected nilpotent Lie group and \(\Gamma\) is a discrete torsion free uniform subgroup. For these spaces we have the additional result that when the homotopy can be accomplished, the resulting \(g\) satisfies \(| \text{fix} (g^ n) |=N(f^ n)\) for all \(n\) with \(N(f^ n) \neq 0\) and if \(\{N(f^ n)\}^ \infty_{n=1} = \{0\}\) then we can choose \(g\) to be periodic point free. Also, when \(f\) is induced by a homomorphism \(F:G \to G\), then we can write \(g=ufv\) where \(u\) and \(v\) are isotopic to the identity. This form for the homotopy is used to find sufficient conditions for \(\text{per} (f) \sim\) finite when \(M\) is a solvmanifold. We then present a model for a specific class of solvmanifolds where these conditions can be considered. This allows us to prove the general result in a variety of low dimensional examples including the Klein bottle.
scientific article; zbMATH DE number 589811