Ergodic theorists and probabilists model physical systems as measure preserving transformations of a measure space. Observables (random variables) on the system correspond to measurable functions on the measure space. The statistical properties of observables under the time-evolution of the system are often determined by whether or not there exist measurable solutions to certain ``cohomological'' equations. It is apriori difficult to use features of the dynamical system such as periodic or homoclinic orbits to rule out the existence of measurable solutions to cohomological equations since such orbits usually have measure zero i.e. they are not seen by the measure-theoretic point of view. We discuss how rigidity theorems enable one in certain contexts to conclude that if there exists a measurable solution then there exists a solution of higher regularity (continuous, Hölder, smooth, ...) which in turn allows one to deduce statistical properties of the system from the periodic or homoclinic data. We give applications to the study of chaotic dynamical systems with Euclidean symmetry.