Expanding maps, Anosov diffeomorphisms and affine structures on infra-nilmanifolds. (Q1399164)

Let \(L\) be a finite dimensional real Lie algebra. An automorphism of \(L\) is called expanding (hyperbolic) if all of its eigenvalues are outside the unit circle (if it has eigenvalues both inside and outside the unit circle, but none on the unit circle). \(L\) admits an expanding (hyperbolic) automorphism if and only if \(L\) admits a positive (mixed, with the zero part equal to zero) \(R\)-grading. It is then shown that any infra-nilmanifold which admits an expanding map or an Anosov diffeomorphism has a complete affine structure.