# Randomness-induced phenomena and negativity of generic random dynamical systems of complex polynomials

We consider random dynamical systems of complex polynomial maps

on the Riemann sphere. In the usual iteration dynamical system

of a single rational map of degree two or more,

we have a kind of chaos in the Julia set and

the Hausdorff dimension of the set of initial values whose Lyapunov

exponent is negative, is positive.

However, in this talk, we show that

for a generic random dynamical systems of complex polynomials,

for all but countable initial values $z$, for almost

every sequence of polynomials,

the Lyapunov exponent is negative and

the chaoticity of the averaged system is much weaker than

usual iteration dynamical system.

There are many new phenomena in random dynamical systems

which cannot hold in the usual iteration dynamical systems.

We call such phenomena "randomness-induced phenomena".

We investigate the mechanisms and background of these phenomena.