  A remarkable feature of fluid dynamics is its relationship with classical
dynamics and statistical mechanics. This has motivated in the past mathematical
investigations concerning, in a special way, the "derivation" based on kinetic
theory, and in particular the Boltzmann equation, of the incompressible
Navier-Stokes equations (INSE). However, the connection determined in this way
is usually merely asymptotic (i.e., it can be reached only for suitable limit
functions) and therefore presents difficulties of its own. This feature has
suggested the search of an alternative approach, based on the construction of a
suitable inverse kinetic theory (IKT; Tessarotto et al., 2004-2007), which can
avoid them. IKT, in fact, permits to achieve an exact representation of the
fluid equations by identifying them with appropriate moment equations of a
suitable (inverse) kinetic equation. The latter can be identified with a
Liouville equation advancing in time a phase-space probability density function
(PDF), in terms of which the complete set of fluid fields (prescribing the
state of the fluid) are determined. In this paper we intend to investigate the
mathematical properties of the underlying \textit{finite-dimensional}
phase-space classical dynamical system, denoted \textit{Navier-Stokes dynamical
system}, which can be established in this way. The result we intend to
establish has fundamental implications both for the mathematical investigation
of Navier-Stokes equations as well as for diverse consequences and applications
in fluid dynamics and applied sciences.
