Spaces of cohomologies associated with linear functional equations

Let F : X → X be a C^k(X), k = [0, ∞], map on a topological space (smooth manifold) X, A : X → End(C^m) and let {Uα} be an F-invariant covering of X. We introduce spaces of cohomologies associated with {Uα} and an operator T = I - R, where (Rφ)(x) = A(x)φ(F(x)) is a weighted substitution operator in C^k(X). This yields a correspondence between Im T and Im T\Uα and the description of Im T in cohomological terms. In particular, it is proven that for any structurally stable diffeomorphism on a circle and for large enough k, the operator T is semi-Fredholm, and a similar result holds for the substitution operators generated by simple multidimensional maps. On the other hand, we show that, in general, the closures of Im T and Im T|Uα are independent.