Minimal sets of periods for torus maps

Abstract

Let $T^r$ be the $r$-dimensional torus, and let $f:T^r\to T^r$ be a map. If $\Per(f)$ denotes the set of periods of $f$, the minimal set of periods of $f$, denoted by $\MPer(f)$, is defined as $\bigcap_{g\cong f}\Per(g)$ where $g:T^r\to T^r$ is homotopic to $f$. First, we characterize the set $\MPer(f)$ in terms of the Nielsen numbers of the iterates of $f$. Second, we distinguish three types of the set $\MPer(f)$ and show that for each type and any given dimension $r$, the variation of $\MPer(f)$ is uniformly bounded in a suitable sense. Finally, we classify all the sets $\MPer(f)$ for self-maps of the $3$-dimensional torus.