This monograph covers the authors' work over the past twenty five years on generalizing the classical results of John Oxtoby and Stan Ulam on the typical dynamical behavior of manifold homeomorphisms which preserve a fixed measure. In the main text of the book we will take a logical rather than historical perspective, designed to give the reader a concise and unified treatment of results we obtained in a series of articles that were written before the overall structure of the theory was clear. However, since the true significance of this field of study can be understood only from a historical perspective, we devote this preface to a discussion of the problem considered by Oxtoby and Ulam when they were Junior Fellows at Harvard in the 1930s, and of their accomplishment in its solution. We shall use their own words where possible.

The origins of Ergodic Theory lie in the study of physical systems which evolve in time as solutions to certain differential equations. Such systems can be initially described by parameters giving the states of the system as points in Euclidean n-space. Taking conservation laws into account, the phase space may be decomposed into lower dimensional manifolds. Regularities in the differential equations obeyed by the system are reflected in the differentiability or the continuity of the flow that describes the evolution of the system over time. Furthermore, Liouville's Theorem ensures that for Hamiltonian systems this flow has an invariant measure.