  We investigate i.i.d. random complex dynamical systems generated by
probability measures on finite unions of the loci of holomorphic families of
rational maps on the Riemann sphere. We show that under certain conditions on
the families, for a generic system, (especially, for a generic random
polynomial dynamical system,) for all but countably many initial values $z$ in
the Riemann sphere, for almost every sequence of maps $\gamma
=(\gamma_{1},\gamma_{2},\ldots )$, the Lyapunov exponent of $\gamma $ at $z$ is
negative. Also, we show that for a generic system, for every initial value $z$
in the Riemann sphere, the orbit of the Dirac measure at $z$ under the
iteration of the dual map of the transition operator tends to a periodic cycle
of measures in the space of probability measures on the Riemann sphere. Note
that these are new phenomena in random complex dynamics which cannot hold in
deterministic complex dynamical systems. We apply the above theory and results
of random complex dynamical systems to finding roots of any polynomials by
random relaxed Newton's methods and we show that for any polynomial $g$, for
any initial value $z$ in the complex plane which is not a root of $g'$, the
random orbit starting with $z$ tends to a root of $g$ almost surely, which is
the virtue of the effect of randomness.
