# Geometric growth for Anosov maps on the $3$ torus

## TL;DR

This paper proves that certain Anosov maps on the 3-torus are smoothly conjugate to their linear parts if their Lyapunov exponents match the geometric growth of invariant foliations, revealing a rigidity property.

## Contribution

It establishes a rigidity result linking Lyapunov exponents and geometric growth to smooth conjugacy for Anosov maps on the 3-torus.

## Key findings

- Lyapunov exponents equal to geometric growth imply smooth conjugacy.
- Rigidity of Anosov maps under specified conditions.
- Characterization of linearizability for 3-torus Anosov maps.

## Abstract

We prove that for Anosov maps of the $3$-torus if the Lyapunov exponents of absolutely continuous measures in every direction are equal to the geometric growth of the invariant foliations then $f$ is $C^1$ conjugated to his linear part.