Constant periodic data and rigidity

Abstract

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, \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, $d\ge 4,$ 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.

arXiv admin note: text overlap with arXiv:1603.06412