On the cohomological equation of a hyperbolic automorphism

Overview & Synopsis

In this paper, we study the discrete cohomological equation (e) : f − f • γ = g associated with a hyperbolic affine automorphism γ : T^d −→ T^d of the torus T d = R^d /Z^d. g being a given C^∞ function and f an unknown function. More precisely, if we denote by E the Fréchet space of all C^∞ functions on T^d and by δ the operator defined on E by δ(f) = f − f • γ, we show that the space δ(E) is a closed subspace of E and we explicitly determine a continuous linear operator L : δ(E) → E such that for every element g of δ(E), the function f = L(g), given by its Fourier series, is a solution of the equation (e).