Constant periodic data and rigidity

In this work we lead with expanding maps of the circle and Anosov diffeomorphisms on ${\mathbb{T}}^d,d\ge 2.$ We prove that, for these maps, $\mathit{\text{constant}}\mathit{\text{periodic}}\mathit{\text{data}}$ 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}$^d, d= 2, 3,$wehaveglobalrigidity\mathrm{.}Inhigherdimensions,$d \geq 4,$ 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.