
In ergodic theory, cohomological questions appear at many places. For example in the construction of some Schroedinger operators (PDF file) with a theorem of Feldman and Moore (PDF file) . 
Maybe the simplest cohomology group is defined for an automorphism T: X > X of a Lebesgue space (X,A,m). The group of measurable sets A (with symmetric difference + as groupoperation) modulo the subgroup { Z = Y + T(Y) } of all coboundaries is the first cohomology group of the Z action with structure group Z _{ 2 }. 
