### 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.

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.

#### Keywords

Anosov diffeomorphism; expanding map; nilpotent Lie group; infra-nilmanifold; affine map

#### Full Text:

FULL TEXT### Refbacks

- There are currently no refbacks.