Affine Anosov Diffeomorphims of Affine Manifolds

̂ M of the connection ∇M to the universal cover ̂ M of M is a locally flat connection. The affine structure of ̂ M is defined by a local diffeomorphism DM : ̂ M → R called the developing map. The developing map gives rise to a representation hM : π1 M → Aff R called the holonomy. The linear part L hM of hM is the linear holonomy. The affine manifold M,∇M is complete if and only if DM is a diffeomorphism. This means also that the connection ∇M is geodesically complete. A diffeomorphism f of M is called an Anosov diffeomorphism f if and only if there exists a norm ‖·‖ on M associated to a differentiable metric 〈·, ·〉, a real number 0 < λ < 1 such that the tangent bundle TM ofM is the direct summand of two bundles TM and TM called, respectively, the stable bundle and the unstable bundle such that