On the Kolmogorov property of a class of infinite measure hyperbolic dynamical systems

Smooth maps with singularities describe important physical phenomena such as the collisions of rigid spheres among them and/or with the walls of a container. Questions about the ergodic properties of these models (which can be mapped into billiard models) were first raised by Boltzmann in the nineteenth century and lie at the foundation of Statistical Mechanics. Billiard models also describe the diffusive motion of electrons bouncing off positive nuclei (Lorentz gas models) and in this situation the physical measure can be considered infinite. It is therefore of great importance to study the ergodic properties of maps when the measure they preserves is infinite. The aim of this thesis is to present an original result on smooth maps with singularities which preserve an infinite measure. Such result establishes the atomicity of the tail $\sigma$-algebra (and hence strong chaotic properties) in the presence of a totally conservative behavior.