Lefschetz numbers for continuous maps, and periods for expanding maps on infra-nilmanifolds

We prove that (1) The Lefschetz number of any continuous map ff on an infra-nilmanifold with holonomy group ΨΨ is L(f)=1|Ψ|∑A∈Ψdet(A∗−f∗)detA∗; (2) The sets of periods for expanding maps on nn-dimensional infra-nilmanifolds are uniformly cofinite, i.e., there is a positive integer m0m0, which depends only on nn, such that for any integer m≥m0m≥m0, for any nn-dimensional infra-nilmanifold MM and for any expanding map ff on MM, there exists a periodic point of ff whose least period is exactly mm. This is a generalization of the main result of [R. Tauraso, Sets of periods for expanding maps on flat manifolds, Monatsh. Math. 128 (1999) 151–157] on flat manifolds to infra-nilmanifolds.