Box dimension of generalized affine fractal interpolation functions (II)

Let $f$ be a generalized affine fractal interpolation function with vertical scaling functions. In this paper, we first estimate $\mathrm{dim}_B \Gamma f$, the box dimension of the graph of $f$, by the sum function of vertical scaling functions. Then we estimate $\mathrm{dim}_B \Gamma f$ by the limits of spectral radii of vertical scaling matrices under certain conditions. As an application, we study the box dimension of the graph of a generalized Weierstrass-type function.Comment: 20 pages, 1 figure. arXiv admin note: text overlap with arXiv:2208.0412

On the existence of cut points of connected generalized Sierpinski carpets

In a previous work joint with Dai and Luo, we show that a connected generalized Sierpinski carpet (or shortly GSCs) has cut points if and only if the associated level-$k$ Hata graph has a long tail for all $k\geq 2$. In this paper, we extend the above result by showing that it suffices to check a finite number of level-$k$ Hata graphs to reach a conclusion. This criterion provides a truly "algorithmic" solution to the cut point problem of connected GSCs. Some interesting examples such as connected GSCs with exactly $n\geq 1$ cut points are also included in addition.Comment: 22 pages, 14 figures. arXiv admin note: text overlap with arXiv:2111.0088