  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.
