What an infra-nilmanifold endomorphism really should be . . .

Karel Dekimpe

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.

Keywords


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

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