Measures of maximal entropy that are SRB

How often does it occur that the measure of maximal entropy of a system is an SRB measure? We study this question for C1+α partially hyperbolic diffeomorphisms isotopic to Anosov (DA-diffeomorphisms) on T³, and establish a rigidity result: the measure of maximal entropy is an SRB measure if and only if the sum of its positive Lyapunov exponents coincides with that of the linear Anosov map A on all periodic orbits of the support of the measure. In that case, the measure is also the unique physical measure. We also show non-Anosov examples satisfying this condition, both in the conservative and in the non-conservative setting. Finally, we prove that a volume-preserving C1+α DA-diffeomorphism on T³ is Anosov if all Lyapunov exponents coincide almost everywhere with those of the linear Anosov in the isotopy class. Consequently, a smooth DA-diffeomorphism is smoothly conjugated to its linear part if and only if all Lyapunov exponents coincide almost everywhere with those of its linear part.