  This work analyses the homogeneous isotropic turbulence by means of the
equivalence between Euler and Lagrange representations of motion, adopting the
bifurcation rates associated with Navier--Stokes and kinematic equations, and
an appropriate hypothesis of fully developed chaos. The equivalence of these
motion descriptions allows to show that kinetic and thermal energy cascade
arise both from the convective term of Liouville equation. Accordingly, these
phenomena, of nondiffusive nature, correspond to a transport in physical space
linked to the trajectories divergence. Both the bifurcation rates are properly
defined, where the kinematic bifurcation rate is shown to be much greater than
Navier--Stokes bifurcation rate. This justifies the proposed hypothesis of
fully developed chaos where velocity field and particles trajectories
fluctuations are statistically uncorrelated. Thereafter, a specific ergodic
property is presented, which relates the statistics of fluid displacement to
that of velocity and temperature fields. A detailed analysis of separation rate
is proposed which studies the statistics of radial velocity component along the
material separation vector. Based on previous elements, the closure formulas of
von K\'arm\'an--Howarth and Corrsin equations are finally achieved. These
closures, of nondiffusive kind, represent a propagation phenomenon, and
coincide with those just presented by the author in previous works,
corroborating the results of these latter.
