  In this work we lead with expanding maps of the circle and Anosov
diffeomorphisms on $\mathbb{T}^d, d \geq 2.$ We prove that, for these maps,
\textit{constant periodic data} imply \textit{same periodic data of these maps
and their linearizations}, so in particular we have smooth conjugacy. For
expanding maps of the circle and Anosov diffeomorphism on $\mathbb{T}^d, d= 2,
3,$ we have global rigidity. In higher dimensions, $d \geq 4,$ we can establish
a result of local rigidity, in several cases. The main tools of this work are
celebrated results of rigidity involving same periodic data with linearization
and results involving topological entropy of a diffeomorphism along an
expanding invariant foliation.
