What an infra-nilmanifold endomorphism really should be

Abstract

Infra-nilmanifold endomorphisms were introduced in the late sixties. They play a very crucial role in dynamics, especially when studying expanding maps and Anosov diffeomorphisms. However, in this note we will explain that the two main results in this area are based on a false result and that although we can repair one of these two theorems, there remains doubt on the correctness of the other one. Moreover, we will also show that the notion of an infra-nilmanifold endomorphism itself has not always been interpreted in the same way. Finally, we define a slightly more general concept of the notion of an infra-nilmanifold endomorphism and explain why this is really the right concept to work with.