General Sierpinski Research Articles

Approximating graphs of a class of general Sierpinski triangles and their normalized Laplacian spectra

The normalized Laplacian spectrum of a graph is an important tool that one can use to find much information about its topological and structural characteristics and also on some relevant dynamical aspects, specifically in relation to random walks. In this paper we devise an essentially algorithm to obtain the approximating graphs of a class of general Sierpinski triangles and their normalized Laplacian spectra, and illustrate such algorithm by a quasi-program of Matlab. In the meantime, our work also enriches the graphs whose spectrum is known.

Hausdorff dimension of asymptotic self-similar sets

In this paper, we introduce the notion of asymptotic self-similar sets on general doubling metric spaces by extending the notion of self-similar sets, and determine their Hausdorff dimensions, which gives an extension of Balogh and Rohner 's result. This is carried out by introducing the notions of almost similarity maps and asymptotic similarity systems. These notions have an advantage of making geometric constructions possible. Actually, as an application, we determined the Hausdorff dimension of general Sierpinski gaskets on complete surfaces constructed by a geometric way in a natural manner.

On the uniqueness of an ergodic measure of full dimension for non-conformal repellers

We give a subclass $\mathcal{L}$ of Non-linear Lalley-Gatzouras carpets and an open set $\mathcal{U}$ in $\mathcal{L}$ such that any carpet in $\mathcal{U}$ has a unique ergodic measure of full dimension. In particular, any Lalley-Gatzouras carpet which is close to a non-trivial general Sierpinski carpet has a unique ergodic measure of full dimension.

Contracting Similarity Fixed Point of General Sierpinski Gasket

The article mainly studies the contracting similarity fixed point and the structure of the general Sierpinski gasket. Firstly, the paper analyzes the importance of contracting similarity fixed point in fractal geometry. Based on a series of definitions, the article studies the contracting similarity fixed point. Then, the paper researches on the structure of the general Sierpinski gasket, and describes it through function. Using a new characteristic function and other techniques, the article gives two important lemmas of the general Sierpinski gasket and the complete proof. By the proved lemmas, the article gets the formula of the contracting similarity fixed point of the Sierpinski gasket, and then proves that the aforementioned contracting similarity fixed points form a new fractal. As application, the paper classifies the above contracting similarity fixed points into three types and gives two examples.

HAUSDORFF DIMENSION OF CERTAIN RANDOM SELF-AFFINE FRACTALS

In this work we are interested in the self-affine fractals studied by Gatzouras and Lalley [5] and by the author [11] who generalize the famous general Sierpinski carpets studied by Bedford [1] and McMullen [13]. We give a formula for the Hausdorff dimension of sets which are randomly generated using a finite number of self-affine transformations each generating a fractal set as mentioned before. The choice of the transformation is random according to a Bernoulli measure. The formula is given in terms of the variational principle for the dimension.

Lipschitz equivalence of a class of general Sierpinski carpets

In this paper, we discuss the Lipschitz equivalence of a class of general Sierpinski carpets in which all non-trivial connected components are line segments. We define a bijection between two link-separated sets with same type by pairing off the basic sets using the indexing by the corresponding symbol space and get a sufficient condition that two general Sierpinski carpets are Lipschitz equivalent. Several examples will be given to illustrate our idea.

A note on the quantization for probability measures with respect to the geometric mean error

We study the quantization with respect to the geometric mean error for probability measures μ on \({\mathbb{R}^d}\) for which there exist some constants C, η > 0 such that \({\mu(B(x,\varepsilon))\leq C\varepsilon^\eta}\) for all e > 0 and all \({x\in\mathbb{R}^d}\) . For such measures μ, we prove that the upper quantization dimension \({\overline{D}(\mu)}\) of μ is bounded from above by its upper packing dimension and the lower one \({\underline{D}(\mu)}\) is bounded from below by its lower Hausdorff dimension. This enables us to calculate the quantization dimension for a large class of probability measures which have nice local behavior, including the self-affine measures on general Sierpinski carpets and self-conformal measures. Moreover, based on our previous work, we prove that the upper and lower quantization coefficient for a self-conformal measure are both positive and finite.

Hausdorff dimension and measure of a class of subsets of the general Sierpinski carpets

In this paper we study a class of subsets of the general Sierpinski carpets for which two groups of allowed digits occur in the expansions with proportional frequency. We calculate the Hausdorff and Box dimensions of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive and finite.

The Hausdorff dimension of sets related to the general Sierpinski carpets

In this paper we study a class of subsets of the general Sierpinski carpets for which the allowed two digits in the expansions occur with proportional frequency. We calculate the Hausdorff and box dimensions of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive finite.

The quantization dimension of the self-affine measures on general Sierpiński carpets

Let μ be a self-affine measure on a general Sierpinski carpet E. We give a characterization for the upper and lower quantization dimension of μ in terms of revised cylinder sets. Using this characterization, we prove that the quantization dimension D r (μ) of μ exists for all r > 0 under an additional condition. We establish an explicit formula for D r (μ) and show that it increases to the box-counting dimension \({dim_B^* \mu}\) of μ as r tends to infinity. For a class of Sierpinski carpets E and the uniform measures μ on E, we show that the quantization dimension of μ coincides with its box-counting dimension and that the D r (μ)-dimensional upper and lower quantization coefficient of μ are both positive and finite.

Measure of full dimension for some nonconformal repellers

Given $(X,T)$ and $(Y,S)$ mixing subshifts of finite type such that $(Y,S)$ is a factor of $(X,T)$ with factor map $\pi$:$\ X\to Y$, and positive Hölder continuous functions $\varphi$:$\ X\to \mathbb{R}$ and $\psi$:$\ Y\to \mathbb{R}$, we prove that the maximum of <p align="center"> $\frac{h_{\mu\circ \pi^{-1}}(S)}{\int \psi\circ\pi\d\mu}+ \frac{h_\mu(T)-h_{\mu\circ \pi^{-1}}(S)}{\int \varphi\d\mu}$ <p align="left" class="times"> over all $T$-invariant Borel probability measures $\mu$ on $X$ is attained on the subset of ergodic measures. Here $h_\mu(T)$ stands for the metric entropy of $T$ with respect to $\mu$. As an application, we prove the existence of an ergodic invariant measure with full dimension for a class of transformations treated in [11], and also for the transformations treated in [17], where the author considers nonlinear skew-product perturbations of <i>general Sierpinski carpets</i>. In order to do so we establish a variational principle for the topological pressure of certain noncompact sets.

SUBSETS OF THE GENERAL SIERPINSKI CARPET WITH MIXED GROUP FREQUENCIES

We consider a class of subsets of the general Sierpinski carpet which is characterized by insisting that the allowed digits in the expansion occur with prescribed mixing group frequencies, determine their Hausdorff dimensions and give the necessary and sufficient conditions for their corresponding Hausdorff measures to be positive and finite.

LIPSCHITZ EQUIVALENCE OF GENERAL SIERPINSKI CARPETS

Recently, a lot of work has been devoted to the study of the Lipschitz equivalence between self-similar sets. Many related results on self-similar sets of the real line have been well-known under the condition that the similarity ratios equal each other. In this paper, we generalize some results to a more general setting. We mainly study the Lipschitz equivalence between two general Sierpinski carpets in ℝ2.

A generalized multifractal spectrum of the general Sierpinski carpets

In this paper we study a class of subsets of the general Sierpinski carpets for which the limiting frequency of a horizontal fibre falls into a prescribed closed interval. We obtain the explicit expression for the Hausdorff dimension of these subsets in terms of the parameters of the construction and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive finite.

A random version of McMullen–Bedford general Sierpinski carpets and its application

We consider a random version of the McMullen–Bedford general Sierpinski carpet which is constructed by randomly choosing patterns in each step instead of a single pattern in its original form. Their Hausdorff, packing and box-counting dimensions are determined. A sufficient condition and a necessary condition for the Hausdorff measures in their dimensions to be positive are given. As an application, we discuss the issue on the intersection of the general Sierpinski carpet with its translations.

Hausdorff dimension of subsets with proportional fibre frequencies of the general Sierpinski carpet

In this paper we study a class of subsets of the general Sierpinski carpets for which the digits in the expansions lie in two specified horizontal fibres with proportional frequencies. We calculate the Hausdorff dimension of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive and finite.

THE MULTIFRACTAL HAUSDORFF AND PACKING MEASURE OF GENERAL SIERPINSKI CARPETS

In this paper, authors study the properties of multifractal Hausdorff and packing measures for a class of self-affine sets and use them to study the multifractal properties of general Sierpinski carpet E, and they get that the multifractal Hausdorff and packing measure are mutual singular, when they are restricted on some subsets of E.

DIMENSIONS OF MEASURE ON GENERAL SIERPINSKI CARPET

Let S=Πi=1∞{0,1,⋯,r-1} and R¯ the general Sierpinski carpet. Let μ be the induced probability measure on R¯ of μ¯ on 5 by Φ, where Φ is the natural surjection from S onto R¯ and μ¯ is the infinite product probability measure corresponding to probability vector (b0,⋯,br-1) with bi=ailognm-1/mα. Authors show that dirmH μ=C_L(μ)=C¯L(μ)=C_(μ)=C¯(μ)=α

Multifractal decomposition of the Sierpinski carpet

We analyse the multifractal decomposition of a measure defined on a general Sierpinski carpet. We compute the dimension spectrum f(α) and we show that this function is real-analytic on a certain domain when it is not degenerated. Actually, we prove that f is the Legendre-Fenchel transformation of a free energy function F which is also real-analytic. These two functions have the typical behaviours. In particular, F is strictly increasing and is in general strictly concave (respectively linear in the degenerate case), and f, on its domain, has a typical shape of a strictly concave function (respectively defined in one point in the degenerate case). We associate also to the singularity sets C α measures which are singular with respect to each other, and we see that these measures are very well fitted to these singularity sets. This work completes with (D. Simpelaere, Chaos, Solitons and Fractals 4(12), 2223–2235 (1994)) the study of the multifractal analysis of the Sierpinski carpets.

The correlation dimension of the Sierpinski carpet

We compute the correlation dimension of a measure defined on a general Sierpinski carpet. We relate this function to a free energy function associated to a partition composed of ‘nearly squares’ and well fitted to the planar Cantor set. Actually, we prove that these functions are real analytic on R , are strictly increasing and are strictly concave (respectively linear in the degenerate case). This is an example of a two-dimensional dynamical system contracting in the two directions with different ratios. We first study measures of Gibbsian type before generalizing to Markovian measures.