Surfaces that undergo random impact of fragments of debris obeying a hierarchy of decreasing sizes are investigated by numerical simulation and analytic solution. The previously introduced hierarchical model for random deposition of debris is extended to allow for periodically varying deposition probabilities P for hills, Q for holes, and spatial rescaling factors lambda. The logarithmic fractal roughness of the surface is found to be robust with respect to these time-dependent perturbations and the model can be solved exactly. The fractal amplitudes, which replace the usual notion of fractal dimension, are periodic functions of the deposition time. Asymptotically, for long times, the amplitudes approach a limit cycle, which can be interpreted as the result of an interaction between the bare amplitudes associated with separate processes with constant deposition parameters. We distinguish amplitude repulsion, attraction, neutrality, coincidence and auto-repulsion. We also solve analytically the transient regime of exponential decay towards asymptotia.