  The foundations for a thermo-statistical description of the called non
extensive Hamiltonian systems are reconsidered. The relevance of the parametric
resonance as a fundamental mechanism of the Hamiltonian chaoticity in those
systems with bound motions in the configurational space is discussed. The
universality of this mechanism suggests us the possibility of performing a
thermo-statistical description with microcanonical basis in the context of the
long-range interacting Hamiltonian systems. The concept of selfsimilarity is
proposed as an appropriate generalization of the well-known extensive
conditions exhibited by the traditional systems, which is used to justify a
given generalized thermodynamic formalism starting from the consideration of
the microcanonical ensemble, i.e. the nonextensive Statistics of Tsallis. These
ideas are illustrated by considering a recent proposed astrophysical model
based on the quasi-ergodic character of the microscopic dynamics of these
paradigmatic examples of real long-range interacting systems.
