Statistical Mechanics has been founded during the XIX-th century by the seminal work of Maxwell, Boltzmann and Gibbs, with the main aim to explain the properties of macroscopic systems from the atomistic point of view. Accordingly, from the very beginning, starting from the Boltzmann's ergodic hypothesis, a basic question was the connection between the dynamics and the statistical properties. This is a rather difficult task and, in spite of the mathematical progress, by Birkhoff and von Neumann, basically ergodic theory had a marginal relevance in the development of the statistical mechanics (at least in the physics community). Partially this was due to a misinterpretation of a result of Fermi and a widely spreaded opinion (based also on the belief of influential scientists as Landau) on the key role of the many degrees of freedom and the practical irrelevance of ergodicity. This point of view found a mathematical support on some results by Khinchin who was able to show that, in systems with a huge number of particles, statistical mechanics works (independently of the ergodicity) just because, on the constant energy surface, the most meaningful physical observables are nearly constant, apart from regions of very small measure,

On the other hand the discovery of the deterministic chaos (from the anticipating work of Poincaré to the contributions, in the second half of the XX-th century, by Chirikov, Hénon, Lorenz and Ruelle, to cite just the most famous) beyond its undoubted relevance for many natural phenomena, showed how the typical statistical features observed in systems with many degrees of freedom, can be generated also by the presence of deterministic chaos in simple systems.