Estimation of errors on perturbation of function contractivity factors and box-counting dimension of hidden variable recurrent fractal interpolation function
Mi-Kyong Ri, Chol-Hui Yun
备注：
arXiv admin note: text overlap with arXiv:1904.11884
In this paper, we study errors on perturbation of function contractivity factors and box-counting dimension of hidden variable recurrent fractal interpolation function (HVRFIF). The HVRFIF is a hidden variable fractal interpolation function (HVFIF) constructed by recurrent iterated function system (RIFS) with function contractivity factors. The contractivity factors of RIFS determine fractal characteristics and shape of its attractor, so that the HVRFIF with function contractivity factors has more flexibility and diversity than the HVFIF with constant contractivity factors. Stability of interpolation function according to perturbation of the contractivity factors and the box-counting dimension of interpolation function plays very important roles in determining whether these functions can be applied to practical problems or not. We first estimate errors on perturbation of function contractivity factors and then obtain the upper and lower bounds of the box-counting dimension of one variable HVRFIF. Finally, in the similar way, we get the lower and upper bounds of box-counting dimension of hidden variable bivariable recurrent fractal interpolation function (HVBRFIF).
英文摘要：
In this paper, we study errors on perturbation of function contractivity factors and box-counting dimension of hidden variable recurrent fractal interpolation function (HVRFIF). The HVRFIF is a hidden variable fractal interpolation function (HVFIF) constructed by recurrent iterated function system (RIFS) with function contractivity factors. The contractivity factors of RIFS determine fractal characteristics and shape of its attractor, so that the HVRFIF with function contractivity factors has more flexibility and diversity than the HVFIF with constant contractivity factors. Stability of interpolation function according to perturbation of the contractivity factors and the box-counting dimension of interpolation function plays very important roles in determining whether these functions can be applied to practical problems or not. We first estimate errors on perturbation of function contractivity factors and then obtain the upper and lower bounds of the box-counting dimension of one variable HVRFIF. Finally, in the similar way, we get the lower and upper bounds of box-counting dimension of hidden variable bivariable recurrent fractal interpolation function (HVBRFIF).
分形插值函数（Fractal interpolation function，简称FIF）是一种特殊类型的连续函数，它对特定的数据集进行插值，对应于该数据集的迭代函数系统（Itered function system，简称IFS）的吸引子就是FIF的图形。凝聚隐变量分形插值函数（CHFIF）是一种既有自仿射性质又有非自仿射性质的广义IFS，它依赖于广义IFS的自由变元和约束自由变元。本文研究了有限个广义数据集的图有向迭代函数系统，证明了吸引子在$\mathbb{R}^{2}$上的投影是CHFIFs插值相应数据集的图.
Dimension Theory of some non-Markovian repellers Part I: A gentle introduction
Michael Barnsley介绍了一类分形集，它们是分段仿射系统的排斥子。对这些分形的研究是由分形图像压缩中出现的某些问题引起的，但我们得到的结果可以应用于计算某些函数的图形的豪斯多夫维数，如广义Takagi函数和分形插值函数。在本文中，我们介绍了这类分形，并提出了工具，在一维动力学和非共形分形理论，需要调查他们。这是一系列两篇论文的第一部分。在后续的工作中，我们将有更多的证明，并应用这里介绍的工具来研究一些分形函数图。
