Spaces of cohomologies associated with linear functional equations

Abstract

Let $F:X\rightarrow X$ be a $C^k(X)$, $k=[0, \infty]$, map on a topological space (smooth manifold) $X$, $A:X\rightarrow \End(C^m)$ and let $\{U_\alpha\}$ be an $F\mbox{-invariant}$ covering of $X$. We introduce spaces of cohomologies associated with $\{U_\alpha\}$ and an operator $T=I - R$, where $(R\phi)(x)=A(x)\phi(F(x))$ is a weighted substitution operator in $C^k(X)$. This yields a correspondence 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 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 $\Im T|U_\alpha$ are independent.