Anosov diffeomorphisms constructed from ${\pi }_k(Diff(S^n))$

We construct Anosov diffeomorphisms on manifolds that are homeomorphic to infranilmanifolds yet have exotic smooth structures. These manifolds are obtained from standard infranilmanifolds by connected summing with certain exotic spheres. Our construction produces Anosov diffeomorphisms of high codimension on infranilmanifolds with irreducible exotic smooth structures.