Entropy rigidity of Anosov flows in dimension three

We show that for a smooth contact Anosov flow on a closed three manifold th e measure of maximal entropy is in the Lebesgue class if and only if the fl ow is, up to finite covers, conjugate to the geodesic flow of a metric of c onstant negative curvature on a closed surface. This shows that the ratio b etween the measure theoretic entropy and the topological entropy of a conta ct Anosov flow is strictly smaller than one on any closed three manifold wh ich is not a Seifert bundle.