Generalized Sierpinski Carpets Research Articles

A Sierpinski carpet like fractal without standard self-similar energy

We construct a Sierpinski carpet like fractal, on which a diffusion with sub-Gaussian heat kernel estimate does not exist, in contrast to previous researches on the existence of such diffusions, on the generalized Sierpinski carpets and recently introduced unconstrained Sierpinski carpets.

Wave Propagation Speed on Fractals

We study the wave propagation speed problem on metric measure spaces, emphasizing on self-similar sets that are not post-critically finite. We prove that a sub-Gaussian lower heat kernel estimate leads to infinite propagation speed, extending a result of Lee (Infinite propagation speed for wave solutions on some p.c.f. fractals, https://archive.org/details/arxiv-1111.2938) to include bounded and unbounded generalized Sierpinski carpets, some fractal blowups, and certain iterated function systems with overlaps. We also formulate conditions under which a Gaussian upper heat kernel estimate leads to finite propagation speed, and apply this result to two classes of iterated function systems with overlaps, including those defining the classical infinite Bernoulli convolutions.

Time Changes of the Brownian Motion: Poincaré Inequality, Heat Kernel Estimate and Protodistance

In this paper, time changes of the Brownian motions on generalized Sierpinski carpets including n n -dimensional cube [ 0 , 1 ] n [0, 1]^n are studied. Intuitively time change corresponds to alteration to density of the medium where the heat flows. In case of the Brownian motion on [ 0 , 1 ] n [0, 1]^n , density of the medium is homogeneous and represented by the Lebesgue measure. Our study includes densities which are singular to the homogeneous one. We establish a rich class of measures called measures having weak exponential decay. This class contains measures which are singular to the homogeneous one such as Liouville measures on [ 0 , 1 ] 2 [0, 1]^2 and self-similar measures. We are going to show the existence of time changed process and associated jointly continuous heat kernel for this class of measures. Furthermore, we obtain diagonal lower and upper estimates of the heat kernel as time tends to 0 0 . In particular, to express the principal part of the lower diagonal heat kernel estimate, we introduce “protodistance”associated with the density as a substitute of ordinary metric. If the density has the volume doubling property with respect to the Euclidean metric, the protodistance is shown to produce metrics under which upper off-diagonal sub-Gaussian heat kernel estimate and lower near diagonal heat kernel estimate will be shown.

Dual graphs and modified Barlow-Bass resistance estimates for repeated barycentric subdivisions

We prove Barlow-Bass type resistance estimates for two random walks associated with repeated barycentric subdivisions of a triangle. If the random walk jumps between the centers of triangles in the subdivision that have common sides, the resistance scales as a power of a constant $ρ$ which is theoretically estimated to be in the interval $5/4≤ρ≤3/2$ , with a numerical estimate $ρ≈1.306$ . This corresponds to the theoretical estimate of spectral dimension $d_S$ between 1.63 and 1.77, with a numerical estimate $d_S≈1.74$ . On the other hand, if the random walk jumps between the corners of triangles in the subdivision, then the resistance scales as a power of a constant $ρ^T = 1/ρ$ , which is theoretically estimated to be in the interval $2/3≤ρ^T≤4/5$ . This corresponds to the spectral dimension between 2.28 and 2.38. The difference between $ρ$ and $ρ^T$ implies that the the limiting behavior of random walks on the repeated barycentric subdivisions is more delicate than on the generalized Sierpinski Carpets, and suggests interesting possibilities for further research, including possible non-uniqueness of self-similar Dirichlet forms.

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.

Topological pressure and dimension estimation

In this paper, we give a survey of recent results in the dimension theory of dynamical systems, with emphasis on the dimension of repellers and hyperbolic sets and the dimension of invariant measures. In the case of conformal dynamics, the theory is completely well understood. However, it still lacks today a satisfactory general approach for the non-conformal case, although a number of interesting and nontrivial developments have been obtained. Indeed, we only well understand some particular non-conformal repellers, e.g., generalized Sierpinski carpets and average conformal repellers.

Log-Periodic Asymptotic Expansion of the Spectral Partition Function for Self-Similar Sets

Let $${{\fancyscript{Z}}(t)}$$ be the partition function (the trace of the heat semigroup) of the canonical Laplacian on a post-critically finite self-similar set (with uniform resistance scaling factor and good geometric symmetry) or on a generalized Sierpinski carpet. It is proved that $${{\fancyscript{Z}}(t)=\sum_{k=0}^{n}t^{-d_{k}/d_{\mathrm{w}}}G_{k}(-\log t) +O\bigl({\rm exp}(-ct^{-\frac{1}{d_{\mathrm{w}}-1}})\bigr)}$$ as $${t\downarrow 0}$$ for some continuous periodic functions $${G_{k}:{\mathbb{R}}\to{\mathbb{R}}}$$ and $${c\in(0,\infty)}$$ . Here $${d_{\mathrm{w}}\in(1,\infty)}$$ denotes the walk dimension, n = 1 for a post-critically finite self-similar set, n = d for a d-dimensional generalized Sierpinski carpet, $${\{d_{k}\}_{k=0}^{n}\subset[0,\infty)}$$ is strictly decreasing with d n = 0, G 0 is strictly positive and G 1 is either strictly positive or strictly negative depending on the (Neumann or Dirichlet) boundary condition.

Self-Assembling Rulers for Approximating Generalized Sierpinski Carpets

Discrete self-similar fractals have been used as test cases for self-assembly, both in the laboratory and in mathematical models, ever since Winfree exhibited a tile assembly system in which the Sierpinski triangle self-assembles. For strict self-assembly, where tiles are not allowed to be placed outside the target structure, it is an open question whether any self-similar fractal can self-assemble. This has motivated the development of techniques to approximate fractals with strict self-assembly. Ideally, such an approximation would produce a structure with the same fractal dimension as the intended fractal, but with specially labeled tiles at positions corresponding to points in the fractal. We show that the Sierpinski carpet, along with an infinite class of related fractals, can approximately self-assemble in this manner. Our construction takes a set of parameters specifying a target fractal and creates a tile assembly system in which the fractal approximately self-assembles. This construction introduces rulers and readers to control the self-assembly of a fractal structure without distorting it. To verify the fractal dimension of the resulting assemblies, we prove a result on the dimension of sets embedded into discrete fractals. We also give a conjecture on the limitations of approximating self-similar fractals.

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.

Asymptotics for Functions Associated with Heat Flow on the Sierpinski Carpet

Abstract We establish the asymptotic behaviour of the partition function, the heat content, the integrated eigenvalue counting function, and, for certain points, the on-diagonal heat kernel of generalized Sierpinski carpets. For all these functions the leading term is of the form xγΦ (log x) for a suitable exponent γ and Φ a periodic function. We also discuss similar results for the heat content of affine nested fractals.

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.

Uniqueness of Brownian motion on Sierpi&#324ski carpets

We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpinski carpet that is invariant with respect to the local symmetries of the carpet. Consequently for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.

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.

Spectral asymptotics for Laplacians on self-similar sets

Given a self-similar Dirichlet form on a self-similar set, we first give an estimate on the asymptotic order of the associated eigenvalue counting function in terms of a ‘geometric counting function’ defined through a family of coverings of the self-similar set naturally associated with the Dirichlet space. Secondly, under (sub-)Gaussian heat kernel upper bound, we prove a detailed short time asymptotic behavior of the partition function, which is the Laplace–Stieltjes transform of the eigenvalue counting function associated with the Dirichlet form. This result can be applicable to a class of infinitely ramified self-similar sets including generalized Sierpinski carpets, and is an extension of the result given recently by B.M. Hambly for the Brownian motion on generalized Sierpinski carpets. Moreover, we also provide a sharp remainder estimate for the short time asymptotic behavior of the partition function.

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.