  On certain self-similar substrates the time behavior of a random walk is
modulated by logarithmic periodic oscillations on all time scales. We show that
if disorder is introduced in a way that self-similarity holds only in average,
the modulating oscillations are washed out but subdiffusion remains as in the
perfect self-similar case. Also, if disorder distribution is appropriately
chosen the oscillations are localized in a selected time interval. Both the
overall random walk exponent and the period of the oscillations are
analytically obtained and confirmed by Monte Carlo simulations.
