Équations cohomologiques de flots riemanniens et de difféomorphismes d’Anosov

Abstract

In this paper: i) We compute the leafwise cohomology of a complete Riemannian Diophantine flow. ii) We solve explicitly the discrete cohomological equation for the Anosov diffeomorphism on the torus $T^n$ defined by a hyperbolic and diagonalizable matrix $A\in \mathit{SL}(n,Z)$ whose eigenvalues are all real positive numbers. We use this to solve the continuous cohomological equation of the Anosov flow $\mathit{F}$ on the hyperbolic torus $T_A^{n+1}$ obtained from $A$ by suspension. This enables us to compute some other geometrical objects associated to the diffeomorphism $A$ and the foliation $\mathit{F}$ like the invariant distributions and the leafwise cohomology.