随机复杂动力学和魔鬼竞技场
Random complex dynamics and devil's coliseums
摘要：
研究了黎曼球面上多项式映射的随机动力学和黎曼球面上多项式映射半群的动力学。特别地，研究了平面后临界集有界的多项式半群$G$的动力学和相应的随机动力学。通常，这种$G$的朱利亚集可以是断开的。我们证明了如果$G$是这样一个半群，那么对于相关的随机动力学，平均系统的混沌在$C^{0}$意义下消失，趋于$\infty$的概率函数$T_{\infty}$在黎曼球面上是Hölder连续的，并且只在$G$的朱利亚集上变化.此外，函数$T_{\infty}$具有某种单调性。事实证明，$T_{\infty}$是一个复杂的模拟魔鬼的楼梯，我们称$T_{\infty}$为“魔鬼的竞技场。“当$G$由两个多项式生成时，我们研究了$T_{\infty}$的细节。在这种情况下，$T_{\infty}$精确地在$G$的朱利亚集上变化，这是一个薄分形集。在此条件下，我们利用几何观测、遍历论、位势论和函数论研究了$T_{\infty}$的点态Hölder指数.特别地，我们证明了对于$G$的朱利亚集上的几乎每一个点$z$，对于不变测度，$T_{\infty}$在$z.$上不可微。我们发现了许多新的随机复动力学现象，这些现象在通常的单个多项式的迭代动力学中是不成立的，并对它们进行了系统的研究。
英文摘要：
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.
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