We investigate i.i.d. random complex dynamical systems generated by probability measures on finite unions of the loci of holomorphic families of rational maps on the Riemann sphere \(\hat{\mathbb {C}}.\) We show that under certain conditions on the families, for a generic system, (especially, for a generic random polynomial dynamical system,) for all but countably many initial values \(z\in \hat{\mathbb {C}}\), for almost every sequence of maps \(\gamma =(\gamma _{1}, \gamma _{2},\ldots )\), the Lyapunov exponent of \(\gamma \) at z is negative. Also, we show that for a generic system, for every initial value \(z\in \hat{\mathbb {C}}\), the orbit of the Dirac measure at z under the iteration of the dual map of the transition operator tends to a periodic cycle of measures in the space of probability measures on \(\hat{\mathbb {C}}\). Note that these are new phenomena in random complex dynamics which cannot hold in deterministic complex dynamical systems. We apply the above theory and results of random complex dynamical systems to finding roots of any polynomial by random relaxed Newton’s methods and we show that for any polynomial g of degree two or more, for any initial value \(z\in \mathbb {C}\) which is not a root of \(g'\), the random orbit starting with z tends to a root of g almost surely, which is the virtue of the effect of randomness.

我们研究在黎曼球面上\（\ hat {\ mathbb {C}}。\）上有理图的全纯族的有位图谱的有限联合的概率测度生成的iid随机复杂动力系统我们证明了在某些条件下这些族，对于一般系统，（尤其是对于一般随机多项式动力系统），对于几乎每一个映射序列\（除了可计数的许多初始值\（z \ in \ hat {\ mathbb {C}} \）而言）\伽马=（\伽马_ {1}，\伽玛_ {2}，\ ldots）\） ，的Lyapunov指数\（\伽马\）在ž是负的。此外，我们表明，对于一般系统，对于每个初始值\（z \ in \ hat {\ mathbb {C}} \），Dirac测度的轨道位于在过渡算子对偶映射的迭代下z趋于\（\ hat {\ mathbb {C}} \）上的概率测度空间中的测度周期性周期。请注意，这些是随机复数动力学中的新现象，无法在确定性复数动力学系统中成立。我们将上述理论和随机复杂动力系统的结果用于通过随机松弛牛顿法找到任何多项式的根，并且我们证明对于任何阶数为2或更大的多项式g，对于任何初始值\（z \ in \ mathbb {C } \）不是\（g'\）的根，以z开头的随机轨道趋于g的根 几乎可以肯定，这是随机效应的优点。