# Constant periodic data and rigidity

## TL;DR

This paper proves that constant periodic data for certain expanding maps and Anosov diffeomorphisms imply smooth conjugacy and rigidity, with results varying between global and local depending on the dimension.

## Contribution

It establishes new rigidity results linking constant periodic data to smooth conjugacy for expanding maps and Anosov diffeomorphisms, extending known results to higher dimensions.

## Key findings

- Constant periodic data implies smooth conjugacy for certain maps.
- Global rigidity results for dimensions 2 and 3.
- Local rigidity results for dimensions 4 and higher.

## Abstract

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.