Pointwise Hoelder exponents of the complex analogues of the Takagi function in random complex dynamics
Pointwise Hoelder exponents of the complex analogues of the Takagi function in random complex dynamics
复制标题
随机复动力学中 Takagi 函数的复类似物的逐点 Hoelder 指数
DOI：
10.1016/j.aim.2017.04.021
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发表时间：
2017
影响因子：
1.7
通讯作者：
Johannes Jaerisch and Hiroki Sumi
中科院分区：
文献类型：
--
作者：
J. Jaerisch and H. Sumi;Johannes Jaerisch and Hiroki Sumi
考虑黎曼球面上具有分离条件和多重极小集的双曲随机复动力系统。我们研究了趋向于一个极小集的概率的函数T的Hölder正则性，T关于概率参数的偏导数可视为Takagi函数的复类，以及T的高阶偏导数C。我们的主要结果给出了T和C的逐点Hölder指数的动态描述，这使得我们可以利用遍历理论中的多重分形形式来确定逐点Hölder指数的谱。此外，我们证明了谱α−的下界严格小于1，这使得我们可以证明平均系统对每个α，1)在α-Hölder连续函数的Banach空间Cα∈(α−上混沌作用，尽管平均系统在Cβ上对小的β和gt；0表现得非常温和(例如我们有谱间隙)。
We consider hyperbolic random complex dynamical systems on the Riemann sphere with separating condition and multiple minimal sets. We investigate the Hölder regularity of the function T of the probability of tending to one minimal set, the partial derivatives of T with respect to the probability parameters, which can be regarded as complex analogues of the Takagi function, and the higher partial derivatives C of T. Our main result gives a dynamical description of the pointwise Hölder exponents of T and C, which allows us to determine the spectrum of pointwise Hölder exponents by employing the multifractal formalism in ergodic theory. Also, we prove that the bottom of the spectrum α− is strictly less than 1, which allows us to show that the averaged system acts chaotically on the Banach space C α of α-Hölder continuous functions for every α∈(α−, 1), though the averaged system behaves very mildly (eg we have spectral gaps) on C β for small β> 0.