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A few years ago, the first example of a closed manifold admitting an Anosovdiffeomorphism but no expanding map was given. Unfortunately, this example isnot explicit and is high-dimensional, although its exact dimension is unknowndue to the type of construction. In this paper, we present a family of concrete12-dimensional nilmanifolds with an Anosov diffeomorphism but no expanding map,where nilmanifolds are defined as the quotient of a 1-connected nilpotent Liegroup by a cocompact lattice. We show that this family has the smallestpossible dimension in the class of infra-nilmanifolds, which is conjectured tobe the only type of manifolds admitting Anosov diffeomorphisms up tohomeomorphism. The proof shows how to construct positive gradings from theeigenvalues of the Anosov diffeomorphism under some additional assumptionsrelated to the rank, using the action of the Galois group on these algebraicunits.

On closed manifolds admitting an Anosov diffeomorphism but no expanding map