The Anosov Relation for Nielsen Numbers of Maps of Infra-Nilmanifolds

A celebrated theorem of Anosov states that for any continuous map f : M → M of a nilmanifold M, one has that N(f) = |L(f)|. In this paper we show that the same holds for any continuous map of an infra-nilmanifold with a holonomy group of odd order. We also establish a necessary and sufficient condition for expanding maps f of infra-nilmanifolds M in order that the Anosov theorem should hold for this specific map. Namely, N(f) = |L(f)| if and only if M is orientable. Finally we introduce the nowhere expanding maps of infra-nilmanifolds for which we show that the Anosov theorem also holds.