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Journal of Fractal Geometry最新文献

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Affine embeddings of Cantor sets on the line 康托集合在直线上的仿射嵌入
IF 0.8 4区 数学
Journal of Fractal Geometry Pub Date : 2016-07-11 DOI: 10.4171/jfg/63
A. Algom
{"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}
引用次数: 11
Completely symmetric resistance forms on the stretched Sierpiński gasket 完全对称的阻力形成在拉伸Sierpiński垫圈上
IF 0.8 4区 数学
Journal of Fractal Geometry Pub Date : 2016-06-28 DOI: 10.4171/JFG/61
Patricia Alonso Ruiz, U. Freiberg, Jun Kigami
{"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}
引用次数: 14
Numerical integration for fractal measures 分形测度的数值积分
IF 0.8 4区 数学
Journal of Fractal Geometry Pub Date : 2016-06-08 DOI: 10.4171/JFG/60
Jens Malmquist, R. Strichartz
{"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}
引用次数: 1
Non-degeneracy of the harmonic structure on Sierpiński gaskets Sierpiński垫片上谐波结构的非简并性
IF 0.8 4区 数学
Journal of Fractal Geometry Pub Date : 2016-05-13 DOI: 10.4171/JFG/73
Konstantinos Tsougkas
{"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}
引用次数: 3
Sobolev algebra counterexamples 索博列夫代数反例
IF 0.8 4区 数学
Journal of Fractal Geometry Pub Date : 2016-05-13 DOI: 10.4171/JFG/59
T. Coulhon, Luke G. Rogers
{"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}
引用次数: 0
Interesting examples in $mathbb{C}^2$ of maps tangent to the identity without domains of attraction $mathbb{C}^2$中的有趣例子是与恒等式相切的无吸引域的映射
IF 0.8 4区 数学
Journal of Fractal Geometry Pub Date : 2016-04-05 DOI: 10.4171/jfg/84
Sara Lapan
{"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}
引用次数: 0
Rigidity and reconstruction for graphs 图的刚性与重构
IF 0.8 4区 数学
Journal of Fractal Geometry Pub Date : 2016-01-29 DOI: 10.4171/JFG/76
G. Cornelissen, J. Kool
{"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}
引用次数: 1
$C^m$ Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of Hausdorff dimension: applications in $mathbb R^1$ Perron-Frobenius算子的特征函数和Hausdorff维数数值计算的新方法:在$mathbb R^1$中的应用
IF 0.8 4区 数学
Journal of Fractal Geometry Pub Date : 2016-01-25 DOI: 10.4171/JFG/62
R. S. Falk, R. Nussbaum
{"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}
引用次数: 13
Non-spectral fractal measures with Fourier frames 傅立叶框架的非谱分形测度
IF 0.8 4区 数学
Journal of Fractal Geometry Pub Date : 2015-09-23 DOI: 10.4171/JFG/52
Chun-Kit Lai, Yang Wang
{"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}
引用次数: 18
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