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Let $f$ be a generalized affine fractal interpolation function with vertical scaling functions. In this paper, we prove the monotonicity of spectral radii of vertical scaling matrices without additional assumptions. We also obtain the irreducibility of these matrices under certain conditions. By these results, we estimate $\mathrm{dim}_B \Gamma f$, the box dimension of the graph of $f$, by the limits of spectral radii of vertical scaling matrices. We also estimate $\mathrm{dim}_B \Gamma f$ directly by the sum function of vertical scaling functions. 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:2208.04126

Let $E$ be the attractor of an iterated function system $\{\phi_i(x)=\rho R_ix+a_i\}_{i=1}^N$ on $\Bbb R^d$, where $0<\rho<1$, $a_i\in \Bbb R^d$ and $R_i$ are orthogonal transformations on $\Bbb R^d$. Suppose that $\{\phi_i\}_{i=1}^N$ satisfies the open set condition, but not the strong separation condition. We show that $E$ can not be generated by any iterated function system of similitudes satisfying the strong separation condition. This gives a partial answer to a folklore question about the separation conditions on the generating iterated function systems of self-similar sets.

In this paper, we study the topological properties and the gap sequences of Bedford-McMullen sets. First, we introduce a topological condition, the component separation condition (CSC), and a geometric condition, the exponential rate condition (ERC). Then we prove that the CSC implies the ERC, and that both of them are sufficient conditions for obtaining the asymptotic estimate of gap sequences. We also explore topological properties of Bedford-McMullen sets and prove that all normal Bedford-McMullen sets with infinitely many connected components satisfy the CSC, from which we obtain the asymptotic estimate of the gap sequences of Bedford-McMullen sets without any restrictions. Finally, we apply our result to Lipschitz equivalence.

In this paper, we present a general framework to construct recurrent fractal interpolation surfaces (RFISs) on rectangular grids. Then we introduce bilinear RFISs, which are easy to be generated while there are no restrictions on interpolation points and vertical scaling factors. We also obtain the box dimension of bilinear RFISs under certain constraints, where the main assumption is that vertical scaling factors have uniform sums under a compatible partition., Comment: 21 pages, 1 figure

In this paper we provide an up-to-date survey on the study of Lipschitz equivalence of self-similar sets. Lipschitz equivalence is an important property in fractal geometry because it preserves many key properties of fractal sets. A fundamental result by Falconer and Marsh [On the Lipschitz equivalence of Cantor sets, \textit{Mathematika}, \textbf{39} (1992), 223--233] establishes conditions for Lipschitz equivalence based on the algebraic properties of the contraction ratios of the self-similar sets. Recently there has been other substantial progress in the field. This paper is a comprehensive survey of the field. It provides a summary of the important and interesting results in the field. In addition we provide detailed discussions on several important techniques that have been used to prove some of the key results. It is our hope that the paper will provide a good overview of major results and techniques, and a friendly entry point for anyone who is interested in studying problems in this field., Comment: 16 pages, 3 figures

Lipschitz equivalence of self-similar sets is an important area in the study of fractal geometry. It is known that two dust-like self-similar sets with the same contraction ratios are always Lipschitz equivalent. However, when self-similar sets have touching structures the problem of Lipschitz equivalence becomes much more challenging and intriguing at the same time. So far the only known results only cover self-similar sets in $\bR$ with no more than 3 branches. In this study we establish results for the Lipschitz equivalence of self-similar sets with touching structures in $\bR$ with arbitrarily many branches. Key to our study is the introduction of a geometric condition for self-similar sets called {\em substitutable}., Comment: 30 pages, 4 figures