  We investigate the classical evolution of a $\phi^4$ scalar field theory,
using in the initial state random field configurations possessing a fractal
measure expressed by a non-integer mass dimension. These configurations
resemble the equilibrium state of a critical scalar condensate. The measures of
the initial fractal behavior vary in time following the mean field motion. We
show that the remnants of the original fractal geometry survive and leave an
imprint in the system time averaged observables, even for large times compared
to the approximate oscillation period of the mean field, determined by the
model parameters. This behavior becomes more transparent in the evolution of a
deterministic Cantor-like scalar field configuration. We extend our study to
the case of two interacting scalar fields, and we find qualitatively similar
results. Therefore, our analysis indicates that the geometrical properties of a
critical system initially at equilibrium could sustain for several periods of
the field oscillations in the phase of non-equilibrium evolution.
