Statistical and Topological Invariants and Ergodicity

At the end of the nineteenth century, first, the works of Poincaré and Boltzmann (1885) came first, followed by those of G.D. Birkhoff and J. Von Neumann in 1931. “The Ergodic hypothesis” is equivalent supposing that the “major part” of the trajectories of a dynamical system is “equally distributed” (on surfaces of constant energy of the phase space) and makes it possible asymptotically to “replace the temporal averages by the spatial averages”.¹ In 1931, the Birkhoff theorem established a rigorous general framework from which the Ergodic theory has been developed with the purpose to study the asymptotic behavior of a dynamical system by means of its invariant measurements (iteration of a transformation, one-parameter flow). The ergodic theory applies to the deterministic case, i.e. dynamical systems defined by differential equations and coupled with the martingale theory, or applied to the probabilistic case of stochastic processes, in particular of the Markovian type.