On closed manifolds admitting an Anosov diffeomorphism but no expanding map

A few years ago, the first example of a closed manifold admitting an Anosov diffeomorphism but no expanding map was given. Unfortunately, this is not explicit and high-dimensional, although its exact dimension unknown due to type construction. In paper, we present family concrete 12-dimensional nilmanifolds with map, where nilmanifold defined as quotient 1-connected nilpotent Lie group by cocompact lattice. We show that has smallest possible in class infra-nilmanifolds, which conjectured be only manifolds diffeomorphisms up homeomorphism. The proof shows how construct positive gradings from eigenvalues under some additional assumptions related rank, using action Galois on these algebraic units.