ABSTRACT
In the century which has elapsed since the creation of the statistical theory of matter, classical Hamiltonian dynamical systems have been among the main test cases for the effectiveness of the concepts of statistical mechanics. In its modern form, the ergodic hypothesis is replaced by the notion of metric transitivity: every subset of a hypersurface of constant energy that is carried into itself by the time development of the system is either of measure zero or is the complement of a subset of measure zero. Poincare’s analysis of the linear approximation to a flow in the neighborhood of a periodic orbit led to a number of conjectures about the possible non-ergodic behavior of the exact flow which were only established in the 1950’s and 1960’s, especially in the work of Kolmogorov, Arnold and Moser. One of the important developments of ergodic theory in the 1930’s was the proof that certain special flows are indeed ergodic.
