  We investigate the dynamics of $2$-generator semigroups of polynomials with
bounded planar postcritical set and associated random dynamics on the Riemann
sphere. Also, we investigate the space ${\cal B}$ of such semigroups. We show
that for a parameter $h$ in the intersection of ${\cal B}$, the hyperbolicity
locus ${\cal H}$ and the closure of the disconnectedness locus (the space of
parameters for which the Julia set is disconnected), the corresponding
semigroup satisfies either the open set condition (and the Bowen's formula) or
that the Julia sets of the two generators coincide. Also, we show that for such
a parameter $h$, if the Julia sets of the two generators do not coincide, then
there exists a neighborhood $U$ of $h$ in the full parameter space ${\cal
P}^{2}$ such that for each parameter in $U$, the Hausdorff dimension of the
Julia set of the corresponding semigroup is strictly less than 2. Moreover, we
show that the intersection of the connectedness locus and ${\cal B}\cap {\cal
H}$ has dense interior. By using the results on the semigroups corresponding to
these parameters, we investigate the associated functions which give the
probability of tending to $\infty$ (complex analogues of the devil's staircase
or Lebesgue's singular functions) and complex analogues of the Takagi function.
