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

2015/10/28 Wed 16:30 - 17:30
Room 110, Building No.3
SUMI Hiroki
Mathmatics School of Science, Osaka University

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.