Local extrema in higher-dimensional toral Anosov systems

Determine whether the SRB entropy functional has no nontrivial local extrema for transitive Anosov systems on tori of dimension greater than two.

Background

For volume-preserving transitive Anosov diffeomorphisms on the two-dimensional torus, the paper cites a result showing that the SRB entropy has neither a local minimum nor a nontrivial local maximum, with its maximum attained at maps smoothly conjugate to automorphisms.

The paper explicitly states that analogous results are not known in dimensions higher than two, leaving open whether the same absence of nontrivial local extrema persists.