Ballistic deposition on deterministic fractals: On the observation of   discrete scale invariance
  The growth of ballistic aggregates on deterministic fractal substrates is studied by means of numerical simulations. First, we attempt the description of the evolving interface of the aggregates by applying the well-established Family-Vicsek dynamic scaling approach. Systematic deviations from that standard scaling law are observed, suggesting that significant scaling corrections have to be introduced in order to achieve a more accurate understanding of the behavior of the interface. Subsequently, we study the internal structure of the growing aggregates that can be rationalized in terms of the scaling behavior of frozen trees, i.e., structures inhibited for further growth, lying below the growing interface. It is shown that the rms height ($h_{s}$) and width ($w_{s}$) of the trees of size $s$ obey power laws of the form $h_{s} \propto s^{\nu_{\parallel}}$ and $w_{s} \propto s^{\nu_{\perp}}$, respectively. Also, the tree-size distribution ($n_{s}$) behaves according to $n_{s}\sim s^{-\tau}$. Here, $\nu_{\parallel}$ and $\nu_{\perp}$ are the correlation length exponents in the directions parallel and perpendicular to the interface, respectively. Also, $\tau$ is a critical exponent. However, due to the interplay between the discrete scale invariance of the underlying fractal substrates and the dynamics of the growing process, all these power laws are modulated by logarithmic periodic oscillations. The fundamental scaling ratios, characteristic of these oscillations, can be linked to the (spatial) fundamental scaling ratio of the underlying fractal by means of relationships involving critical exponents. 
Claudio M. Horowitz, Federico Roma and Ezequiel V. Albano