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Ergodic theory studies dynamical systems in spaces with an invariant measure. In particular, the long term limit of the average of the position of a particle is studied. In statistical physics, the derivation of various thermodynamic principles depends much on what is called the ergodic hypothesis (wikipedia) saying that in long the term in a system with a large number of particles the probability of passing on the constant energy surface near some particular state is independent of the state on that surface and its probability does not depend on the initial conditions (other than energy).
Ergodic reasoning is important also in construction of counterexamples in geometry and geometric group theory as well as in the construction and study of various measures, representations, random walks and so on related to algebraic objects and objects of harmonic analysis.
A list of references available at this course page.
Discussion of cohomology-considerations in ergodic theory is in
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