  The stability against perturbations of a dynamical system conserving a
generalized phase-space volume is studied by exploiting the similarity between
statistical physics formalism and that of ergodic theory. A general continuity
theorem is proved. Resulting from this theorem the double average - time and
ensemble - of an observable of a weakly perturbed ergodic dynamical system is
only slightly changed, even in the infinite time limit. Consequences of this
statistical analogue of the structural stability are: extension of the range of
practical applicability of the Boltzmann Ergodic Hypothesis, justification of
the perturbation method in statistical physics, justification of the numerical
approximations in molecular dynamic calculations and smoothness of the
transition from bounded to unbounded motion as observed in numerical simulation
of anomalous transport in tokamaks.
