##
Periodic Points of Anosov Diffeomorphisms

by Thomas F. Banchoff and Michael I. Rosen

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.