  This paper continues the study of the structure of finite intersections of
general multiplicative translates
$\mathcal{C}(M_1,\ldots,M_n)=\frac{1}{M_1}\Sigma_{3,\bar{2}}\cap\cdots\cap\frac{1}{M_n}\Sigma_{3,\bar{2}}$
for integers $1\leq M_1<\cdots<M_n$, where $\Sigma_{3,\bar{2}}$ denotes the
$3$-adic Cantor set. This study was motivated by questions concerning the
discrete dynamical system on the $3$-adic integers $\mathbb{Z}_3$ given by
multiplication by $2$. The exceptional set $\mathcal{E}(\mathbb{Z}_3)$ is
defined to be the set of all elements of $\mathbb{Z}_3$ whose forward orbits
under this action intersect the $3$-adic Cantor set $\Sigma_{3,\bar{2}}$
infinitely many times. It is conjectured that it has Hausdorff dimension $0$.
Part I showed that upper bounds on the Hausdorff dimension of the exceptional
set can be extracted from knowing Hausdorff dimensions of sets of the kind
above, in cases where all $M_i$ are powers of $2$. These intersection sets were
shown to be fractals whose points have $3$-adic expansions describable by
labeled paths in a finite automaton, whose Hausdorff dimension is exactly
computable and is of the form $\log_3(\beta)$ where $\beta$ is a real algebraic
integer. It gave algorithms for determination of the automaton, and computed
examples showing that the dependence of the automaton and the value $\beta$ on
the parameters $(M_1,\ldots,M_n)$ is complicated. The present paper studies two
new infinite families of examples, illustrating interesting behavior of the
automata and of the Hausdorff dimension of the associated fractals. One family
has associated automata whose directed graph has a nested sequence of strongly
connected components of arbitrarily large depth. The second family leads to an
improved upper bound for the Hausdorff dimension of $\mathcal{E}(\mathbb{Z}_3)$
of $\log_3 \phi \approx 0.438018$, where $\phi$ denotes the Golden ratio.
