  The dynamic and kinetic behavior of processes occurring in fractals with
spatial discrete scale invariance (DSI) is considered. Spatial DSI implies the
existence of a fundamental scaling ratio (b_1). We address time-dependent
physical processes, which as a consequence of the time evolution develop a
characteristic length of the form $\xi \propto t^{1/z}$, where z is the dynamic
exponent. So, we conjecture that the interplay between the physical process and
the symmetry properties of the fractal leads to the occurrence of time DSI
evidenced by soft log-periodic modulations of physical observables, with a
fundamental time scaling ratio given by $\tau = b_1 ^z$. The conjecture is
tested numerically for random walks, and representative systems of broad
universality classes in the fields of irreversible and equilibrium critical
phenomena.
