Dynamical systems and systems of equations

Random Dynamical Systems of Regular Polynomial Maps on $\Bbb{C}{2}$

speaker: Hiroki Sumi (Kyoto University)

abstract: We introduce the notion of mean stability in i.i.d. random (holomorphic) 2-dimensional dynamical systems.We can see that a generic random dynamical system of regular polynomial maps on $\Bbb{P}^{2}$ (the complex 2-dimensional projective space) having an attractor in the line at infinity is mean stable. If a random holomorphic dynamical system on $\Bbb{P}^{2}$ is mean stable then for each $z$ in $\Bbb{P}^{2}$, for a.e. orbit starting with $z$, the Lyapunov exponent is negative. Moreover, if a random holomorphic dynamical system on $\Bbb{P}^{2}$ is mean stable, then for any $z$ in $\Bbb{P}^{2}$, the orbit of the Dirac measure at $z$ under the iterations of the dual map of the transition operator converges to a periodic cycle of probability measures.Note that the above statement cannot hold for deterministic dynamics of a single regular polynomial map $f$ of degree two or more. We see many randomness-induced phenomena (phenomena in random dynamical systems which cannot hold for iteration dynamics of single maps).In this talk, we see randomness-induced order.

timetable:
Thu 23 Jun, 11:20 - 12:20, Aula Dini
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