  We study the effects that the constant periodic data condition have on
topological entropy of Anosov diffeomorphisms. Under constant periodic data
condition we prove that Anosov diffeomorphism has finitely many measures of
maximal entropy and each one of them is absolutely continuous with respect to
Lebesgue. From this, in the setting of $C^{\infty}-$Anosov diffeomorphisms
satisfying constant periodic data, we provide a characterization of
transitivity property via measures of maximal entropy.
