Smale has examined both toral diffeomorphisms (introduced by Thom) and nontoral Anosov diffeomorphisms defined on nilmanifolds in a recent article. Smale presents two arguments demonstrating that the periodic points of toral diffeomorphisms are dense. (One is based on the generalized Birkhoff theorem for dynamical systems, and the other relies on an algebraic argument.
This article gives a direct and elementary proof of a slightly stronger result:
Theorem 1. Every point with all coordinates rational is a periodic point of any toral diffeomorphism. Moreover, for almost all toral diffeomorphisms, all periodic point are rational.
Part 2 extends this result to one of the examples of a nontoral diffeomorphism in Smale's article, thus showing the analogy with the toral case. The third, more formal part of the paper gives a proof of the theorem in full generality for the class of Anosov diffeomorphisms on nilmanifolds.