Recent Contributions to the Ergodic Theory

I. The Original Formulations.-Early in the kinetic theory of matter it was recognized that the replacement of mean values in time by mean values in phase-space is an indispensable operation.In order to justify this substitution, recourse was had by Boltzmann and Maxwell to their celebrated Ergodic Hypothesis, which states that, in the systems considered in the kinetic theory, each particular motion will pass, sooner or later, through every state of motion, or phase (combination of position and velocity), consistent with its energy; and (implicitly) that the length of time during which it exhibits a given set of phases is proportional to the relative abundance of the latter.In other words, each path-curve in phase- space passes through every point of its energy hypersurface, remaining in a given region of the latter for a length of time proportional to the extent (hypervolume) of the region.It was recognized by Boltzmann and Maxwell that account must be taken of all possible single-valued integrals of ordinary type, thus reducing the dimensionality of the phase-space.The extreme unlikelihood of the Ergodic Hypothesis, owing to the presence of special periodic motions, was pointed out by Kelvin.'Its mathematical impossibility was apparent to Poincar6,2 who indicated the only possible direction of effective modification, namely, the later Quasi-Ergodic Hypothesis of P. and T. Ehrenfest:3Let Q be the hypersurface in phase-space corresponding to the given value of the energy of the system, and suppose its extent to be finite.Then each non-exceptional path-curve in Q passes through every region of Q of positive volume, remaining there for an average time equal to the ratio of this volume to that of U.Furthermore, P. and T. Ehrenfest4 observed that even if the system is quasi-ergodic in the sense that everywhere dense path-curves exist, the mean time of sojourn along the path-curve through P in a given region M may vary discontinuously from path to path.For the detailed history of the qualifications and speculations, the reader is referred to the "Encyklopadie der Mathematischen Wissenschaften."5II.Introduction of the Modern Theory ofReal Variables into Dynamics.-In his discussion of recurrent motion, H. Poincar66 introduces the funda- mental notion of a dynamical property which, without being true for all possible motions, has a probability of one of being realized.Poincare wrote before Lebesgue's great work, but the very steps of his proof, as well as the formulation of his theorem, are all in almost an exact form for