The miracle of entropy - that the entropy of a measure preserving transformation calculated forward in time (for T) and backwards in time (for T^{-1}) are equal - is, depending on point of view and the definition used, either a triviality or highly surprising. Entropy theory (of Z-actions) plays a key role in analyzing the rigidity of algebraic (diagonalizable) Z^k actions; I show how a strong version of this symmetry property of entropy is useful in studying the measure classification question for such actions.
Joint work with Manfred Einsiedler.