SPACES OF COHOMOLOGIES ASSOCIATED WITH LINEAR FUNCTIONAL-EQUATIONS

Let F : X --> X be a C-k(X), k = [0, infinity], map on a topological s pace (smooth manifold) X, A : X --> End(C-m) and let {U-alpha} be an F -invariant covering of X. We introduce spaces of cohomologies associat ed with {U-alpha} and an operator T = I - R, where (R phi)(x) = A(x)ph i(F(x)) is a weighted substitution operator in C-k(X). This yields a c orrespondence between Im T and Im T/U-alpha and the description of Im T in cohomological terms. In particular, it is proven that for any str ucturally stable diffeomorphism on a circle and for large enough k, th e operator T is semi-Fredholm, and a similar result holds for the subs titution operators generated by simple multidimensional maps. On the o ther hand, we show that, in general, the closures of Im T and Im T/U-a lpha are independent.