Resumen de On the cohomological equation of a linear contraction

Régis Leclercq, Abdellatif Zeggar

• In this paper, we study the discrete cohomological equation of a contracting linear automorphism A of the Euclidean space Rd. More precisely, if δ is the cobord operator defined on the Fréchet space E = Cl (Rd) (0 ≤ l ≤ ∞) by: δ(h) = h − h ◦ A, we show that:

  If E = C0(Rd), the range δ (E) of δ has infinite codimension and its closure is the hyperplane E0 consisting of the elements of E vanishing at 0. Consequently, H1 (A, E) is infinite dimensional non Hausdorff topological vector space and then the automorphism A is not cohomologically C0-stable.

  If E = Cl (Rd), with 1 ≤ l ≤ ∞, the space δ (E) coincides with the closed hyperplane E0. Consequently, the cohomology space H1 (A, E) is of dimension 1 and the automorphism A is cohomologically Cl-stable.