  The goal of this paper is to construct invariant dynamical objects for a (not
necessarily invertible) smooth self map of a compact manifold. We prove a
result that takes advantage of differences in rates of expansion in the terms
of a sheaf cohomological long exact sequence to create unique lifts of finite
dimensional invariant subspaces of one term of the sequence to invariant
subspaces of the preceding term. This allows us to take invariant cohomological
classes and under the right circumstances construct unique currents of a given
type, including unique measures of a given type, that represent those classes
and are invariant under pullback. A dynamically interesting self map may have a
plethora of invariant measures, so the uniquess of the constructed currents is
important. It means that if local growth is not too big compared to the growth
rate of the cohomological class then the expanding cohomological class gives
sufficient "marching orders" to the system to prohibit the formation of any
other such invariant current of the same type (say from some local dynamical
subsystem). Because we use subsheaves of the sheaf of currents we give
conditions under which a subsheaf will have the same cohomology as the sheaf
containing it. Using a smoothing argument this allows us to show that the sheaf
cohomology of the currents under consideration can be canonically identified
with the deRham cohomology groups. Our main theorem can be applied in both the
smooth and holomorphic setting.

Dynamical Systems