  We investigate the dynamics of forward or backward self-similar systems
(iterated function systems) and the topological structure of their invariant
sets. We define a new cohomology theory (interaction cohomology) for forward or
backward self-similar systems. We show that under certain conditions, the space
of connected components of the invariant set is isomorphic to the inverse limit
of the spaces of connected components of the realizations of the nerves of
finite coverings${\cal U} $ of the invariant set, where each ${\cal U}$
consists of (backward)images of the invariant set under elements of finite word
length. Inparticular, we give a criterion for the invariant set to be
connected. Moreover, we give a sufficient condition for the first cohomology
group to have infinite rank. As an application, we obtain many results on the
dynamics of finitely generated semigroups of polynomials on the complex plane.
Moreover, we define postunbranched systems and we investigate the interaction
cohomology groups of such systems. Many examples are given.
