Linear functional-differential equations on the line

Multidimensional equation Sigma(j=1)(l)(D(j)y)(F(j)x) = gamma(x) is considered. Here x is an element of R-1 and D-j : C-infinity(R-1,C-m) --> C-infinity(R-1,C-n) are given differential linear operators. We build a theory of this equation via "common dynamics" of maps F-j. In particular, it is proved that the operator T-0 : C-infinity(R-1,C-m) --> C-infinity(R-1,C-n) with affine transformations F(j)x = alpha(j)x + beta(j) and differential linear operators D-j with constant coefficients is semi-Fredholm.