On the cohomological equation of a linear contraction


  • Régis Leclercq Université Polytechnique Hauts-de-France.
  • Abdellatif Zeggar Université Polytechnique Hauts-de-France.




fréchet space, cohomological equation


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.

Author Biographies

Régis Leclercq, Université Polytechnique Hauts-de-France.

Laboratoire CERAMATHS, INSA Hauts-de-France.

Abdellatif Zeggar, Université Polytechnique Hauts-de-France.

Laboratoire CERAMATHS, INSA Hauts-de-France.


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How to Cite

R. Leclercq and A. Zeggar, “On the cohomological equation of a linear contraction”, Proyecciones (Antofagasta, On line), vol. 41, no. 5, pp. 1075-1091, Sep. 2022.




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