  We investigate the dynamics of polynomial semigroups (semigroups generated by
a family of polynomial maps on the Riemann sphere) and the random dynamics of
polynomials on the Riemann sphere. Combining the dynamics of semigroups and the
fiberwise (random) dynamics, we give a classification of polynomial semigroups
$G$ such that $G$ is generated by a compact family $\Gamma $, the planar
postcritical set of $G$ is bounded, and $G$ is (semi-) hyperbolic. In one of
the classes, we have that for almost every sequence $\gamma \in \Gamma
^{\Bbb{N}}$, the Julia set $J_{\gamma}$ of $\gamma $ is a Jordan curve but not
a quasicircle, the unbounded component of the Fatou set $F_{\gamma}$ of
$\gamma$ is a John domain, and the bounded component of $F_{\gamma}$ is not a
John domain. Note that this phenomenon does not hold in the usual iteration of
a single polynomial. Moreover, we consider the dynamics of polynomial
semigroups $G$ such that the planar postcritical set of $G$ is bounded and the
Julia set is disconnected. Those phenomena of polynomial semigroups and random
dynamics of polynomials that do not occur in the usual dynamics of polynomials
are systematically investigated.
