Expanding maps, Anosov diffeomorphisms and affine structures on infra-nilmanifolds

In this paper we present a complete description of Lie algebras admitting an expanding and/or a hyperbolic automorphism in terms of R-gradings of these Lie algebras. Using this, we then show that any infra-nilmanifold which admits an expanding map or an Anosov diffeomorphism also has a complete, affinely flat structure.