广义仿射分形插值函数的Assouad型维数及其应用
Assouad type dimensions of generalized affine fractal interpolation functions and their applications
- National Institute of Technology Rourkela（印度国家技术学院鲁尔基拉分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究广义仿射分形函数图像的Assouad谱与维数，建立上下界并给出显式表达式，应用于Weierstrass和Takagi函数，解决了Fraser提出的开放问题。
AI中文摘要：
本文研究了由广义仿射迭代函数系统生成的分形函数图像的Assouad谱和Assouad维数。我们利用广义仿射构造中的缩放函数和底层划分，建立了Assouad谱的上下界。若缩放函数是Lipschitz连续的且划分是均匀的，我们得到了相关图像Assouad谱的显式表达式。这些结果建立了定义广义仿射分形函数的参数与其图像局部多尺度几何之间的联系。作为应用，我们考虑了广义仿射分形函数的两个经典例子，即Weierstrass函数和Takagi函数。对于经典Weierstrass函数$W$，其图像$\Gamma_W$的盒维数为$2+\log_N\lambda$，我们得到$\dim_A^\theta(\Gamma_W) \leq \frac{2+\log_N\lambda-\theta}{1-\theta}$，其中$\theta\in \left(0,\log_N\frac{1}{\lambda}\right)$。若$\lambda^2 N <1$，我们得到$\dim_A(\Gamma_W)\geq 1 +\log_N\left(\frac{1}{\lambda}\right)$。对于经典Takagi函数$T$，其图像为$\Gamma_T$，我们证明$\dim_A^\theta(\Gamma_T)=1$，其中$\theta\in(0,1)$，因此其拟Assouad维数等于$1$。这些结果解决了Fraser提出的关于图像维数的开放问题。
英文摘要：
In this article, we investigate the Assouad spectrum and Assouad dimension of graphs of fractal functions generated by generalized affine iterated function systems. We establish upper and lower bounds for the Assouad spectrum in terms of the scaling functions and the underlying partition of the generalized affine construction. If the scaling function is Lipschitz continuous and the partition is uniform, we obtain an explicit expression for the Assouad spectrum of the associated graph. These results provide a connection between the parameters defining the generalized affine fractal function and the local multiscale geometry of its graph. As applications, we consider two classic examples of generalized affine fractal functions, namely the Weierstrass and Takagi functions. For the classical Weierstrass function $W$, whose graph $Γ_W$ has the box dimension $2+\log_Nλ$, we obtain \[ \dim_A^θ(Γ_W) \leq \frac{2+\log_Nλ-θ}{1-θ}, \qquad θ\in \left(0,\log_N\frac{1}λ\right). \] and if $λ^2 N <1$, we get \[ \dim_A(Γ_W)\geq 1 +\log_N\left(\frac{1}λ\right). \] For the classical Takagi function $T$ with graph $Γ_T$, we show that \[ \dim_A^θ(Γ_T)=1, θ\in(0,1), \] and consequently its quasi-Assouad dimension is equal to $1.$ These results settle an open problem on dimension of graphs posed by Fraser.
