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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 $f$ be a generalized affine fractal interpolation function with vertical scaling function $S$. In this paper, we study $\dim_B \Gamma f$, the box dimension of the graph of $f$, under the assumption that $S$ is a Lipschtz function. By introducing vertical scaling matrices, we estimate the upper bound and the lower bound of oscillations of $f$. As a result, we obtain explicit formula of $\dim_B \Gamma f$ under certain constraint conditions., Comment: 18 pages, 1 figure

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.

Let $(K_1,\ldots,K_n)$ be a Cantor type graph-directed attractors in $\mathbb{R}^d$. By creating an auxiliary graph, we provide an effective criterion of whether $K_i\cap K_j$ is empty for every pair of $1\leq i,j\leq n$. Moreover, the emptiness can be checked by examining only a finite number of the attractor's geometric approximations. The method is also applicable for more general graph-directed systems. As an application, we are able to determine the connectedness of all $d$-dimensional generalized Sierpi\'nski sponges of which the corresponding IFSs are allowed to have rotational and reflectional components., Comment: 24 pages, 4 figures

In a previous work joint with Dai and Luo, we show that a connected generalized Sierpi\'nski carpet (or shortly a GSC) has cut points if and only if the associated $n$-th Hata graph has a long tail for all $n\geq 2$. In this paper, we extend the above result by showing that it suffices to check a finite number of those graphs to reach a conclusion. This criterion provides a truly "algorithmic" solution to the cut point problem of connected GSCs. We also construct for each $m\geq 1$ a connected GSC with exactly $m$ cut points and demonstrate that when $m\geq 2$, such a GSC must be of the so-called fragile type., Comment: 28 pages, 7 figures

We investigate a homeomorphism problem on a class of self-similar sets called generalized Sierpinski carpets (or shortly GSCs). It follows from two well-known results by Hata and Whyburn that a connected GSC is homeomorphic to the standard Sierpinski carpet if and only if it has no local cut points. On the one hand, we show that to determine whether a given GSC is connected, it suffices to iterate the initial pattern twice. On the other hand, we obtain two criteria: (1) for a connected GSC to have cut points, (2) for a connected GSC with no cut points to have local cut points. With these two criteria, we characterize all GSCs that are homeomorphic to the standard Sierpinski carpet. Our results on cut points and local cut points hold for Baranski carpets, too. Moreover, we extend the connectedness result to Baranski sponges. Thus, we also characterize when a Baranski carpet is homeomorphic to the standard GSC., Comment: 41 pages, 14 figures

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

We construct certain metrics on the Sierpinski carpet via a class of self-similar weight functions. Using these metrics and by applying known results, we obtain the two-sided sub-Gaussian heat kernel estimates of time change of the standard diffusion on the Sierpinski carpet with respect to self-similar measures. This proves a conjecture by Kigami., Comment: 21 pages, 4 figures

In the paper, we provide an effective method for the Lipschitz equivalence of two-branch Cantor sets and three-branch Cantor sets by studying the irreducibility of polynomials. We also find that any two Cantor sets are Lipschitz equivalent if and only if their contraction vectors are equivalent provided one of the contraction vectors is homogeneous., Comment: 12 pages

In this paper, we first characterize the finiteness of fractal interpolation functions (FIFs) on post critical finite self-similar sets. Then we study the Laplacian of FIFs with uniform vertical scaling factors on Sierpinski gasket (SG). As an application, we prove that the solution of the following Dirichlet problem on SG is an FIF with uniform vertical scaling factor $\frac{1}{5}$: $\Delta u=0$ on $SG\setminus \{q_1,q_2,q_3\}$, and $u(q_i)=a_i$, $i=1,2,3$, where $q_i$, $i=1,2,3$, are boundary points of SG., Comment: 10 papges