# Equation cohomologique d'un automorphisme affine hyperbolique du tore

## TL;DR

This paper investigates the cohomological equation for hyperbolic affine automorphisms of the torus, demonstrating the closedness of the image of the cobord operator and the existence of a continuous solution operator.

## Contribution

It establishes the closedness of the cobord operator’s image and constructs a continuous linear solution operator for the cohomological equation in this setting.

## Key findings

- The image of the cobord operator is a closed subspace of the function space.
- The cohomology space is a nontrivial Fréchet space.
- A continuous linear operator solving the cohomological equation exists.

## Abstract

We study the discrete cohomological equation of a hyperbolic affine automorphism Y of the torus (whose linear part is not necessarily diagonalisable). More precisely ; if d is the cobord operator defined by : d (h) = h-hoY for every element h of the Fr\'echet space E of the differentiable functions on the torus, we show that the image Im(d) is a closed of E and that consequently the space of cohomology H^1(Y; E):=E/Im(d) is a nontrivial Fr\'echet space. We also prove the existence of a continuous linear operator L defined from Im(d) to E such that for every element g of Im(d), the image f = L(g) is a solution of the discrete cohomological equation f-foY = g.