Let $sin (0,1)$, and let $Fsubset mathbb{R}$ be a self similar set such that $0 0$ such that if $F$ admits an affine embedding into a homogeneous self similar set $E$ and $0 leq dim_H E - dim_H F < delta$ then (under some mild conditions on $E$ and $F$) the contraction ratios of $E$ and $F$ are logarithmically commensurable. This provides more evidence for a Conjecture of Feng, Huang, and Rao, that states that these contraction ratios are logarithmically commensurable whenever $F$ admits an affine embedding into $E$ (under some mild conditions). Our method is a combination of an argument based on the approach of Feng, Huang, and Rao, with a new result by Hochman, which is related to the increase of entropy of measures under convolutions.
设$sin (0,1)$和$Fsubset mathbb{R}$是一个自相似集，使得$0 0$这样，如果$F$允许仿射嵌入到同质的自相似集$E$和$0 leq dim_H E - dim_H F < delta$中，那么(在$E$和$F$的一些温和条件下)$E$和$F$的收缩比是对数可通约的。这为Feng, Huang和Rao的猜想提供了更多的证据，该猜想指出，只要$F$允许仿射嵌入$E$(在一些温和的条件下)，这些收缩比率在对数上是可通约的。我们的方法是基于Feng, Huang和Rao的方法的论证与Hochman的新结果的结合，该结果与卷积下测度熵的增加有关。
{"title":"Affine embeddings of Cantor sets on the line","authors":"A. Algom","doi":"10.4171/jfg/63","DOIUrl":"https://doi.org/10.4171/jfg/63","url":null,"abstract":"Let $sin (0,1)$, and let $Fsubset mathbb{R}$ be a self similar set such that $0 0$ such that if $F$ admits an affine embedding into a homogeneous self similar set $E$ and $0 leq dim_H E - dim_H F < delta$ then (under some mild conditions on $E$ and $F$) the contraction ratios of $E$ and $F$ are logarithmically commensurable. This provides more evidence for a Conjecture of Feng, Huang, and Rao, that states that these contraction ratios are logarithmically commensurable whenever $F$ admits an affine embedding into $E$ (under some mild conditions). Our method is a combination of an argument based on the approach of Feng, Huang, and Rao, with a new result by Hochman, which is related to the increase of entropy of measures under convolutions.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2016-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/jfg/63","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70871364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The stretched Sierpinski gasket, SSG for short, is the space obtained by replacing every branching point of the Sierpinski gasket by an interval. It has also been called "deformed Sierpinski gasket" or "Hanoi attractor". As a result, it is the closure of a countable union of intervals and one might expect that a diffusion on SSG is essentially a kind of gluing of the Brownian motions on the intervals. In fact, there have been several works in this direction. There still remains, however, "reminiscence" of the Sierpinski gasket in the geometric structure of SSG and the same should therefore be expected for diffusions. This paper shows that this is the case. In this work, we identify all the completely symmetric resistance forms on SSG. A completely symmetric resistance form is a resistance form whose restriction to every contractive copy of SSG in itself is invariant under all geometrical symmetries of the copy, which constitute the symmetry group of the triangle. We prove that completely symmetric resistance forms on SSG can be sums of the Dirichlet integrals on the intervals with some particular weights, or a linear combination of a resistance form of the former kind and the standard resistance form on the Sierpinski gasket.
{"title":"Completely symmetric resistance forms on the stretched Sierpiński gasket","authors":"Patricia Alonso Ruiz, U. Freiberg, Jun Kigami","doi":"10.4171/JFG/61","DOIUrl":"https://doi.org/10.4171/JFG/61","url":null,"abstract":"The stretched Sierpinski gasket, SSG for short, is the space obtained by replacing every branching point of the Sierpinski gasket by an interval. It has also been called \"deformed Sierpinski gasket\" or \"Hanoi attractor\". As a result, it is the closure of a countable union of intervals and one might expect that a diffusion on SSG is essentially a kind of gluing of the Brownian motions on the intervals. In fact, there have been several works in this direction. There still remains, however, \"reminiscence\" of the Sierpinski gasket in the geometric structure of SSG and the same should therefore be expected for diffusions. This paper shows that this is the case. In this work, we identify all the completely symmetric resistance forms on SSG. A completely symmetric resistance form is a resistance form whose restriction to every contractive copy of SSG in itself is invariant under all geometrical symmetries of the copy, which constitute the symmetry group of the triangle. We prove that completely symmetric resistance forms on SSG can be sums of the Dirichlet integrals on the intervals with some particular weights, or a linear combination of a resistance form of the former kind and the standard resistance form on the Sierpinski gasket.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2016-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JFG/61","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70871223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We find estimates for the error in replacing an integral $int f dmu$ with respect to a fractal measure $mu$ with a discrete sum $sum_{x in E} w(x) f(x)$ over a given sample set $E$ with weights $w$. Our model is the classical Koksma-Hlawka theorem for integrals over rectangles, where the error is estimated by a product of a discrepancy that depends only on the geometry of the sample set and weights, and variance that depends only on the smoothness of $f$. We deal with p.c.f self-similar fractals, on which Kigami has constructed notions of energy and Laplacian. We develop generic results where we take the variance to be either the energy of $f$ or the $L^1$ norm of $Delta f$, and we show how to find the corresponding discrepancies for each variance. We work out the details for a number of interesting examples of sample sets for the Sierpinski gasket, both for the standard self-similar measure and energy measures, and for other fractals.
我们发现在用一个给定样本集$E$上的一个权重为$w$的离散和$sum_{x in E} w(x) f(x)$代替一个积分$int f dmu$关于一个分形测度$mu$的误差估计。我们的模型是用于矩形积分的经典Koksma-Hlawka定理，其中误差通过仅取决于样本集和权重的几何形状的差异和仅取决于$f$平滑度的方差的乘积来估计。我们处理p.c.f自相似分形，Kigami在其上构造了能量和拉普拉斯的概念。我们开发了通用结果，其中我们将方差作为$f$的能量或$Delta f$的$L^1$范数，并且我们展示了如何找到每个方差的相应差异。我们为Sierpinski垫片的一些有趣的样本集例子，包括标准的自相似度量和能量度量，以及其他分形计算出了细节。
{"title":"Numerical integration for fractal measures","authors":"Jens Malmquist, R. Strichartz","doi":"10.4171/JFG/60","DOIUrl":"https://doi.org/10.4171/JFG/60","url":null,"abstract":"We find estimates for the error in replacing an integral $int f dmu$ with respect to a fractal measure $mu$ with a discrete sum $sum_{x in E} w(x) f(x)$ over a given sample set $E$ with weights $w$. Our model is the classical Koksma-Hlawka theorem for integrals over rectangles, where the error is estimated by a product of a discrepancy that depends only on the geometry of the sample set and weights, and variance that depends only on the smoothness of $f$. We deal with p.c.f self-similar fractals, on which Kigami has constructed notions of energy and Laplacian. We develop generic results where we take the variance to be either the energy of $f$ or the $L^1$ norm of $Delta f$, and we show how to find the corresponding discrepancies for each variance. We work out the details for a number of interesting examples of sample sets for the Sierpinski gasket, both for the standard self-similar measure and energy measures, and for other fractals.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2016-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JFG/60","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70871505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the harmonic extension matrices for the level-k Sierpinski Gasket are invertible for every k>2. This has been previously conjectured to be true by Hino in [6] and [7] and tested numerically for k<50. We also give a necessary condition for the non-degeneracy of the harmonic structure for general finitely ramified self-similar sets based on the vertex connectivity of their first graph approximation.
{"title":"Non-degeneracy of the harmonic structure on Sierpiński gaskets","authors":"Konstantinos Tsougkas","doi":"10.4171/JFG/73","DOIUrl":"https://doi.org/10.4171/JFG/73","url":null,"abstract":"We prove that the harmonic extension matrices for the level-k Sierpinski Gasket are invertible for every k>2. This has been previously conjectured to be true by Hino in [6] and [7] and tested numerically for k<50. We also give a necessary condition for the non-degeneracy of the harmonic structure for general finitely ramified self-similar sets based on the vertex connectivity of their first graph approximation.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2016-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JFG/73","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70871781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the Euclidean setting the Sobolev spaces $W^{alpha,p}cap L^infty$ are algebras for the pointwise product when $alpha>0$ and $pin(1,infty)$. This property has recently been extended to a variety of geometric settings. We produce a class of fractal examples where it fails for a wide range of the indices $alpha,p$.
{"title":"Sobolev algebra counterexamples","authors":"T. Coulhon, Luke G. Rogers","doi":"10.4171/JFG/59","DOIUrl":"https://doi.org/10.4171/JFG/59","url":null,"abstract":"In the Euclidean setting the Sobolev spaces $W^{alpha,p}cap L^infty$ are algebras for the pointwise product when $alpha>0$ and $pin(1,infty)$. This property has recently been extended to a variety of geometric settings. We produce a class of fractal examples where it fails for a wide range of the indices $alpha,p$.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2016-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JFG/59","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70871445","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give an interesting example of a map in $mathbb{C}^2$ that is tangent to the identity, but that does not have a domain of attraction along any of its characteristic direction. This map has three characteristic directions, two of which are not attracting while the third attracts points to that direction, but not to the origin. In addition, we show that if we add higher degree terms to this map, sometimes a domain of attraction along one of its characteristic directions will exist and sometimes one will not.
{"title":"Interesting examples in $mathbb{C}^2$ of maps tangent to the identity without domains of attraction","authors":"Sara Lapan","doi":"10.4171/jfg/84","DOIUrl":"https://doi.org/10.4171/jfg/84","url":null,"abstract":"We give an interesting example of a map in $mathbb{C}^2$ that is tangent to the identity, but that does not have a domain of attraction along any of its characteristic direction. This map has three characteristic directions, two of which are not attracting while the third attracts points to that direction, but not to the origin. In addition, we show that if we add higher degree terms to this map, sometimes a domain of attraction along one of its characteristic directions will exist and sometimes one will not.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2016-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70871931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present measure theoretic rigidity for graphs of first Betti number b>1 in terms of measures on the boundary of a 2b-regular tree, that we make explicit in terms of the edge-adjacency and closed-walk structure of the graph. We prove that edge-reconstruction of the entire graph is equivalent to that of the "closed walk lengths".
{"title":"Rigidity and reconstruction for graphs","authors":"G. Cornelissen, J. Kool","doi":"10.4171/JFG/76","DOIUrl":"https://doi.org/10.4171/JFG/76","url":null,"abstract":"We present measure theoretic rigidity for graphs of first Betti number b>1 in terms of measures on the boundary of a 2b-regular tree, that we make explicit in terms of the edge-adjacency and closed-walk structure of the graph. We prove that edge-reconstruction of the entire graph is equivalent to that of the \"closed walk lengths\".","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2016-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JFG/76","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70871856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In the one dimensional case, our methods require only C^3 regularity of the maps in the IFS. The key idea, which has been known in varying degrees of generality for many years, is to associate to the IFS a parametrized family of positive, linear, Perron-Frobenius operators L_s. The operators L_s can typically be studied in many different Banach spaces. Here, unlike most of the literature, we study L_s in a Banach space of real-valued, C^k functions, k >= 2; and we note that L_s is not compact, but has a strictly positive eigenfunction v_s with positive eigenvalue lambda_s equal to the spectral radius of L_s. Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value s=s_* for which lambda_s =1. This eigenvalue problem is then approximated by a collocation method using continuous piecewise linear functions (in one dimension) or bilinear functions (in two dimensions). Using the theory of positive linear operators and explicit a priori bounds on the derivatives of the strictly positive eigenfunction v_s, we give rigorous upper and lower bounds for the Hausdorff dimension s_*, and these bounds converge to s_* as the mesh size approaches zero.
本文提出了一种计算迭代函数系统不变集的Hausdorff维数的新方法。在一维情况下，我们的方法只需要IFS中映射的C^3正则性。多年来，人们在不同程度上已经知道了它的关键思想，就是将一个正的、线性的、Perron-Frobenius算子L_s的参数化族与IFS联系起来。通常可以在许多不同的巴拿赫空间中研究算子L_s。这里，与大多数文献不同，我们研究了实值C^k函数的Banach空间中的L_s, k >= 2;我们注意到L_s不是紧化的，但有一个严格正的特征函数v_s，其正特征值lambda_s等于L_s的谱半径。在对IFS的适当假设下，IFS不变集的Hausdorff维数为值s=s_*，其中lambda_s =1。然后用连续分段线性函数(一维)或双线性函数(二维)的配置方法近似该特征值问题。利用正线性算子理论和严格正特征函数v_s导数的显式先验界，给出了严格的Hausdorff维s_*的上界和下界，并且当网格尺寸趋近于0时，这些上界收敛于s_*。
{"title":"$C^m$ Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of Hausdorff dimension: applications in $mathbb R^1$","authors":"R. S. Falk, R. Nussbaum","doi":"10.4171/JFG/62","DOIUrl":"https://doi.org/10.4171/JFG/62","url":null,"abstract":"We develop a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In the one dimensional case, our methods require only C^3 regularity of the maps in the IFS. The key idea, which has been known in varying degrees of generality for many years, is to associate to the IFS a parametrized family of positive, linear, Perron-Frobenius operators L_s. The operators L_s can typically be studied in many different Banach spaces. Here, unlike most of the literature, we study L_s in a Banach space of real-valued, C^k functions, k >= 2; and we note that L_s is not compact, but has a strictly positive eigenfunction v_s with positive eigenvalue lambda_s equal to the spectral radius of L_s. Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value s=s_* for which lambda_s =1. This eigenvalue problem is then approximated by a collocation method using continuous piecewise linear functions (in one dimension) or bilinear functions (in two dimensions). Using the theory of positive linear operators and explicit a priori bounds on the derivatives of the strictly positive eigenfunction v_s, we give rigorous upper and lower bounds for the Hausdorff dimension s_*, and these bounds converge to s_* as the mesh size approaches zero.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2016-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JFG/62","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70871282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We generalize the compatible tower condition given by Strichartz to the almost-Parseval-frame tower and show that non-trivial examples of almost-Parseval-frame tower exist. By doing so, we demonstrate the first singular fractal measure which has only finitely many mutually orthogonal exponentials (and hence it does not admit any exponential orthonormal bases), but it still admits Fourier frames.
{"title":"Non-spectral fractal measures with Fourier frames","authors":"Chun-Kit Lai, Yang Wang","doi":"10.4171/JFG/52","DOIUrl":"https://doi.org/10.4171/JFG/52","url":null,"abstract":"We generalize the compatible tower condition given by Strichartz to the almost-Parseval-frame tower and show that non-trivial examples of almost-Parseval-frame tower exist. By doing so, we demonstrate the first singular fractal measure which has only finitely many mutually orthogonal exponentials (and hence it does not admit any exponential orthonormal bases), but it still admits Fourier frames.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2015-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JFG/52","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70871430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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