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We investigate the dynamics of $2$-generator semigroups of polynomials withbounded planar postcritical set and associated random dynamics on the Riemannsphere. Also, we investigate the space ${\cal B}$ of such semigroups. We showthat for a parameter $h$ in the intersection of ${\cal B}$, the hyperbolicitylocus ${\cal H}$ and the closure of the disconnectedness locus (the space ofparameters for which the Julia set is disconnected), the correspondingsemigroup satisfies either the open set condition (and the Bowen's formula) orthat the Julia sets of the two generators coincide. Also, we show that for sucha parameter $h$, if the Julia sets of the two generators do not coincide, thenthere exists a neighborhood $U$ of $h$ in the full parameter space ${\calP}^{2}$ such that for each parameter in $U$, the Hausdorff dimension of theJulia set of the corresponding semigroup is strictly less than 2. Moreover, weshow that the intersection of the connectedness locus and ${\cal B}\cap {\calH}$ has dense interior. By using the results on the semigroups corresponding tothese parameters, we investigate the associated functions which give theprobability of tending to $\infty$ (complex analogues of the devil's staircaseor Lebesgue's singular functions) and complex analogues of the Takagi function.

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