Cooperation principle, stability and bifurcation in random complex dynamics
Cooperation principle, stability and bifurcation in random complex dynamics
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随机复动力学中的合作原理、稳定性和分岔
DOI：
10.1016/j.aim.2013.05.023
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
H. Sumi
中科院分区：
文献类型：
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作者：
I. Kim;C. Lecuire and K. Ohshika;Ken'ichi Ohshika;H. Sumi;H. Sumi
本文研究了黎曼球面上有理映射的随机动力学和有理映射半群的动力学。我们表明，对于多项式的随机复动力学，一般而言，由于发电机的自动合作，平均系统的混沌消失在任何点在C。本文研究了作用在C ∞上的（Hölder）连续函数空间上的转移算子的迭代和谱性质。我们还研究了随机复动力学的稳定性和分岔。我们证明了在多项式随机动力系统空间中稳定系统的集合是开的和稠密的。证明了对于一个稳定的系统，只存在n个极小集，每个极小集都是吸引的，且C ∞上的Hölder连续函数在过渡算子作用下的轨道指数快速地趋向于过渡算子的酉特征向量的有限线性组合的有限维空间U.结合线性算子的扰动理论，我们得到，对于由有限族有理映射构成的稳定系统，到空间U的投影依赖于概率参数的实分析。通过对趋于极小集的概率函数关于概率参数的偏导数，我们引入了一个新概念，即Takagi函数的复模拟。
We investigate the random dynamics of rational maps and the dynamics of semigroups of rational maps on the Riemann sphere C ˆ. We show that regarding random complex dynamics of polynomials, generically, the chaos of the averaged system disappears at any point in C ˆ, due to the automatic cooperation of the generators. We investigate the iteration and spectral properties of transition operators acting on the space of (Hölder) continuous functions on C ˆ. We also investigate the stability and bifurcation of random complex dynamics. We show that the set of stable systems is open and dense in the space of random dynamical systems of polynomials. Moreover, we prove that for a stable system, there exist only finitely many minimal sets, each minimal set is attracting, and the orbit of a Hölder continuous function on C ˆ under the transition operator tends exponentially fast to the finite-dimensional space U of finite linear combinations of unitary eigenvectors of the transition operator. Combining this with the perturbation theory for linear operators, we obtain that for a stable system constructed by a finite family of rational maps, the projection to the space U depends real-analytically on the probability parameters. By taking a partial derivative of the function of probability of tending to a minimal set with respect to a probability parameter, we introduce a complex analogue of the Takagi function, which is a new concept.