Irreversibility and extended formulation of classical and quantum nonintegrable dynamics

One of the basic problems in modern physics is the elucidation of the time paradox. The traditional formulation of laws of nature makes no distinction between past and future. The formulation of laws of physics that include time symmetry breaking has now been realized for classes of dynamical systems such as Hamiltonian chaotic maps and large Poincare systems (LPS). For these systems, we have derived a complex irreducible spectral representation for the operators associated to their time evolution (such as the Perron-Frobenius operator or the Liouville operator). This thesis provides extensive theoretical and numerical studies to prove and construct such an extended formalism. For scattering systems, classical or quantum, the limitation of trajectory and wave function formulation toward "persistent interaction" makes inevitable the reconsideration of the eigenvalue problem of the Liouville operator involving singular distribution functions. The resulting complex spectral representation is natural for thermodynamic systems and is applied to Lorentz gas. The compatibility between dissipation and dynamical causality is retained in this statistical formulation of dynamics that includes irreversibility.