The space of $2$-generator postcritically bounded polynomial semigroups and random complex dynamics

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.