Minimal sets of periods for torus maps
Let \(f:X\to X\) be a self-map of a compact connected polyhedron \(X\), and \(n\) be a natural number. For \(\text{Fix} (f)\) and \(P_n(f)\) we denote \[ \text{Fix}(f)= \{x\in X\mid x=f(x)\}, \quad P_n(f)= \{x\in X\mid x= f^n(x), \text{ but }x\neq f^k(x),\;k< n\}. \] Denote by \(\text{Per}(f)= \{n\in N\mid P_n(f)\neq 0\}\) and by \(M \operatorname {Per}(f)= \bigcap \operatorname {Per}(g)\), where the intersection is over all \(g\) which are homotopy equivalent to \(f\). The present paper deals with the problem of determining the set of periods of the periodic orbits of a map given the homotopy class of the map. The authors characterize the set \(M\operatorname {Per}(f)\) in terms of Nielsen numbers of the iterates of \(f\). Furthermore the authors classify all the sets \(M\operatorname {Per}(f)\) for self-maps of the 3D torus.
scientific article; zbMATH DE number 1492521