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 (b1). We address time-dependent physical processes which, as a consequence of the time evolution, develop a characteristic length of the form ξ∝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 τ=b1z. The conjecture is tested numerically for single random walks, annihilating random walks, and representative systems of broad universality classes in the fields of irreversible and equilibrium critical phenomena.Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas