The Abel equation and total solvability of linear functional equations

We investigate the solvability in continuous functions of the Abel equation Mathematical bold italic small phi sign(Fx) - Mathematical bold italic small phi sign(x) = 1 where is a given continuous mapping of a topological space X. This property depends on the dynamics generated by F. The solvability of all linear equations P(x)ψ(Fx) + Q(x)ψ(x) = γ(x) follows from solvability of the Abel equation in case F is a homeomorphism. If F is noninvertible but X is locally compact then such a total solvability is determined by the same property of the cohomological equation Mathematical bold italic small phi sign(Fx) - Mathematical bold italic small phi sign(x) = γ(x). The smooth situation can also be considered in this way.