  Statistical mechanics can only be ultimately justified in terms of
microscopic dynamics (classical, quantum, relativistic, or any other). It is
known that Boltzmann-Gibbs statistics is based on the hypothesis of exponential
sensitivity to the initial conditions, mixing and ergodicity in Gibbs
$\Gamma$-space. What are the corresponding hypothesis for nonextensive
statistical mechanics? A scenario for answering such question is advanced,
which naturally includes the {\it a priori} determination of the entropic index
$q$, as well as its cause and manifestations, for say many-body Hamiltonian
systems, in (i) sensitivity to the initial conditions in Gibbs $\Gamma$-space,
(ii) relaxation of macroscopic quantities towards their values in anomalous
stationary states that differ from the usual thermal equilibrium (e.g., in some
classes of metastable or quasi-stationary states), and (iii) energy
distribution in the $\Gamma$-space for the same anomalous stationary states.
