Measures of maximal entropy that are SRB

Abstract: A smooth conservative DA-diffeomorphism is smoothly conjugated to its Anosov linear part if and only if all Lyapunov exponents coincide almost everywhere with those of its linear part. A more general result for entropy maximizing measures of $C^{1+\alpha}$ partially hyperbolic diffeomorphisms isotopic to Anosov (DA-diffeomorphisms) on $T^3$ is that they are SRB measures if and only if the sum of its positive Lyapunov exponents coincides with that of the linear Anosov map on all periodic orbits of the support of the measure. In that case, the measure is also the unique physical measure. This rigidity result is not as strong as in the A. Katok rigidity conjecture. Examples are provided.