Random complex dynamics and devil's coliseums

Abstract

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 Hölder 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 Hölder 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.