Diffusion approximation and hyperbolic automorphisms of the torus

In this article a diffusion equation is obtained as a limit of a reversible kinetic equation scaled appropriately. This limiting diffusion is produced by the collisions of the particles with the boundary. Indeed, these particles follow a reversible reflection law having convenient mixing properties. This model, based on "Arnold's cat map", can be handled with Fourier series instead of the symbolic dynamics associated to a Markov partition. As a consequence, optimal convergence results can be obtained by elementary means and illustrate the apparition of irreversibility in macroscopic limits.