For a normal subgroup $N$ of the free group $\F_{d}$ with at least two
generators we introduce the radial limit set $L_{r}(N,\Phi)$ of $N$ with
respect to a graph directed Markov system $\Phi$ associated to $\F_{d}$. These
sets are shown to provide fractal models of radial limit sets of normal
subgroups of Kleinian groups of Schottky type. If $\Phi$ is a symmetric linear
graph directed Markov system associated to $\F_{d}$ and $N$ is a normal
subgroup of $\F_{d}$, then we show for the Hausdorff dimension $\dim_{H}$ of
the two associated radial limit sets that we have
$\dim_{H}(L_{r}(N,\Phi))=\dim_{H}(L_{r}(\F_{d}))$ if and only if the quotient
group $\F_{d}/N$ is amenable. This extends a result of Brooks for normal
subgroups of Kleinian groups to a large class of fractal sets. Moreover, we
show that if $\F_{d}/N$ is non-amenable then
$\dim_{H}(L_{r}(N,\Phi))>\dim_{H}(L_{r}(\F_{d},\Phi))/2$. This extends results
by Falk and Stratmann and by Roblin.