  We describe a new statistical pattern in the chaotic dynamics of closed
inflationary cosmologies, associated with the partition of the Hamiltonian
rotational motion energy and hyperbolic motion energy pieces, in a linear
neighborhood of the saddle-center present in the phase space of the models. The
hyperbolic energy of orbits visiting a neighborhood of the saddle-center has a
random distribution with respect to the ensemble of initial conditions, but the
associated histograms define a statistical distribution law of the form $p(x) =
C x^{-\gamma}$, for almost the whole range of hyperbolic energies considered.
We present numerical evidence that $\gamma$ determines the dimension of the
fractal basin boundaries in the ensemble of initial conditions. This
distribution is universal in the sense that it does not depend on the
parameters of the models and is scale invariant. We discuss possible physical
consequences of this universality for the physics of inflation.tribution law of
the form $p(x) = C x^{-\gamma}$, for almost the whole range of hyperbolic
energies considered. We present numerical evidence that $\gamma$ determines the
dimension of the fractal basin boundaries in the ensemble of initial
conditions. This distribution is universal in the sense that it does not depend
on the parameters of the models and is scale invariant. We discuss possible
physical consequences of this universality for the physics of inflation.
