arXiv subjects

Huo-Jun Ruan

Publications and source records attributed to Huo-Jun Ruan.

16 recordsLinked to original sources

arXiv2025

Box dimension of the graphs of recurrent fractal interpolation functions

Let $f$ be a generalized affine recurrent fractal interpolation function with vertical scaling functions. In this paper, by introducing underlying local iterated function systems of $f$, we define restricted vertical scaling matrices. Then we prove the monotonicity of spectral radii of these matrices without additional conditions. We also prove the irreducibility of these matrices under the assumption that vertical scaling functions are positive. With these results, we estimate the upper and lower box dimensions of the graphs of $f$ by the limits of spectral radii of restricted vertical scaling matrices. In particular, we obtain an explicit formula of the box dimension of the graph of $f$ under certain constraint conditions.

math.CA↗

arXiv2025

From Lipschitz embedding to Lipschitz equivalence between dust-like self-similar sets

Let $K,F\subset\mathbb{R}^d$ be two dust-like self-similar sets sharing the same Hausdorff dimension. We consider when the mere existence of a Lipschitz embedding from $K$ to $F$ already implies their Lipschitz equivalence. Our main result is threefold: (1) if the Lipschitz image of $K$ intersects $F$ in a set of positive Hausdorff measure, then $K$ admits a Lipschitz surjection onto $F$; (2) if $F$ is in addition homogeneous, then the generating iterated function systems of $K, F$ should have algebraically dependent ratios and consequently, $K$ and $F$ are Lipschitz equivalent; (3) the Lipschitz equivalence can fail without the homogeneity assumption. This answers two questions in Balka and Keleti [Adv. Math. 446 (2024), 109669].

math.CA↗

arXiv2025

The connectedness of Sierpiński sponges with rotational and reflectional components and associated graph-directed systems

We provide two methods to characterize the connectedness of all $d$-dimensional generalized Sierpiński sponges whose corresponding IFSs are allowed to have rotational and reflectional components. Our approach is to reduce it to an intersection problem between the coordinates of graph-directed attractors. More precisely, let $(K_1,\ldots,K_n)$ be a Cantor-type graph-directed attractor in $\mathbb{R}^d$. By creating an auxiliary graph, we provide an effective criterion for 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 geometric approximations of the attractor. The approach is also applicable to more general graph-directed systems.

math.GN↗

arXiv2024

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 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 $\mathrm{dim}_B Γf$, the box dimension of the graph of $f$, by the limits of spectral radii of vertical scaling matrices. We also estimate $\mathrm{dim}_B Γf$ 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.

math.CA↗

arXiv2022

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 Sierpiński 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.

math.GN↗

arXiv2022

Connectedness and local cut points of generalized Sierpinski carpets

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.

math.GN↗

arXiv2022

A separation property for iterated function systems of similitudes

Let $E$ be the attractor of an iterated function system $\{ϕ_i(x)=ρR_ix+a_i\}_{i=1}^N$ on $\Bbb R^d$, where $0<ρ<1$, $a_i\in \Bbb R^d$ and $R_i$ are orthogonal transformations on $\Bbb R^d$. Suppose that $\{ϕ_i\}_{i=1}^N$ 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.

math.DS↗

arXiv2022

Box dimension of generalized affine fractal interpolation functions

Let $f$ be a generalized affine fractal interpolation function with vertical scaling function $S$. In this paper, we study $\dim_B Γf$, 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 $\dim_B Γf$ under certain constraint conditions.

math.MG↗

arXiv2022

Gap sequences and Topological properties of Bedford-McMullen 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.

math-ph↗

arXiv2020

Construction and box dimension of recurrent fractal interpolation surfaces

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.

math.MG↗

arXiv2018

Metrics on the Sierpinski carpet by weight functions

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.

math.FA↗

arXiv2017

Lipschitz equivalence of Cantor sets and irreducibility of polynomials

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.

math.GT↗

arXiv2016

Energy and Laplacian of Fractal Interpolation Functions

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 $\frac{1}{5}$: $Δu=0$ 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.

math.FA↗

arXiv2013

Lipschitz Equivalence of Self-Similar Sets: Algebraic and Geometric Properties

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.

math.MG↗

arXiv2012

Lipschitz equivalence of self-similar sets with touching structures

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}.

math.MG↗

arXiv2009

The resolvent kernel for PCF self-similar fractals

For the Laplacian $Δ$ defined on a p.c.f. self-similar fractal, we give an explicit formula for the resolvent kernel of the Laplacian with Dirichlet boundary conditions, and also with Neumann boundary conditions. That is, we construct a symmetric function $G^{(λ)}$ which solves $(λ\mathbb{I} - Δ)^{-1} f(x) = \int G^{(λ)}(x,y) f(y) dμ(y)$. The method is similar to Kigami's construction of the Green kernel in \cite[\S3.5]{Kig01} and is expressed as a sum of scaled and "translated" copies of a certain function $ψ^{(λ)}$ which may be considered as a fundamental solution of the resolvent equation. Examples of the explicit resolvent kernel formula are given for the unit interval, standard Sierpinski gasket, and the level-3 Sierpinski gasket $SG_3$.

math.AP↗