We are interested in the quantum dynamics of systems for which either the geometry or the spectrum is fractal. We focus here on the case of a triadic Cantor set energy spectrum. Due to the discrete scaling symmetry inherent to fractal structures, we expect to observe, for some relevant physical quantities, a typical behavior, characterized by a power law modulated by log-periodic oscillations. More specifically, we derive an analytical expression of the mean return probability, and discuss two other quantities : the participation ratio and the RMS displacement. This study is motivated by recent experiments on cavity polaritons placed in a Fibonacci potential, whose spectrum is known to be Cantor like.