  We investigate the random dynamics of polynomial maps on the Riemann sphere
and the dynamics of semigroups of polynomial maps on the Riemann sphere. In
particular, the dynamics of a semigroup $G$ of polynomials whose planar
postcritical set is bounded and the associated random dynamics are studied. In
general, the Julia set of such a $G$ may be disconnected. We show that if $G$
is such a semigroup, then regarding the associated random dynamics, the chaos
of the averaged system disappears in the $C^{0}$ sense, and the function
$T_{\infty}$ of probability of tending to $\infty$ is H\"{o}lder continuous on
the Riemann sphere and varies only on the Julia set of $G$. Moreover, the
function $T_{\infty}$ has a kind of monotonicity. It turns out that
$T_{\infty}$ is a complex analogue of the devil's staircase, and we call
$T_{\infty}$ a "devil's coliseum." We investigate the details of $T_{\infty}$
when $G$ is generated by two polynomials. In this case, $T_{\infty}$ varies
precisely on the Julia set of $G$, which is a thin fractal set. Moreover, under
this condition, we investigate the pointwise H\"{o}lder exponents of
$T_{\infty}$ by using some geometric observations, ergodic theory, potential
theory and function theory. In particular, we show that for almost every point
$z$ in the Julia set of $G$ with respect to an invariant measure, $T_{\infty}$
is not differentiable at $z.$ We find many new phenomena of random complex
dynamics which cannot hold in the usual iteration dynamics of a single
polynomial, and we systematically investigate them.
