Uta Freiberg, Nico Heizmann, Robin Kaiser, Ecaterina Sava-Huss
We consider the doubly infinite Sierpiński gasket graph $mathsf{SG}_0$, rescale it by factor $2^{-n}$, and on the rescaled graphs $mathsf{SG}_n=2^{-n}mathsf{SG}0$, for every $ninmathbb{N}$, we investigate the limit shape of three aggregation models with initial configuration $sigma_n$ of particles supported on multiple vertices. The models under consideration are: divisible sandpile in which the excess mass is distributed among the vertices until each vertex is stable and has mass less or equal to one, internal DLA in which particles do random walks until finding an empty site, and rotor aggregation in which particles perform deterministic counterparts of random walks until finding an empty site. We denote by $mathsf{SG}=text{cl}(bigcup{n=0}^{infty}mathsf{SG}_n)$ the infinite Sierpiński gasket, which is a closed subset of $mathbb{R}^2$, for which $mathsf{SG}_n$ represents the level-$n$ approximating graph, and we consider a continuous function $sigmacolon mathsf{SG}tomathbb{N}$. For $sigma$ we solve the obstacle problem, and we describe the noncoincidence set $Dsubset mathsf{SG}$ as the solution of a free boundary problem on the fractal $mathsf{SG}$. If the discrete particle configurations $sigma_n$ on the approximating graphs $mathsf{SG}_n$ converge pointwise to the continuous function $sigma$ on the limit set $mathsf{SG}$, we prove that, as $ntoinfty$, the scaling limits of the three aforementioned models on $mathsf{SG}_n$ starting with initial particle configuration $sigma_n$ converge to the deterministic solution $D$ of the free boundary problem on the limit set $mathsf{SG}subsetmathbb{R}^2$. For $D$ we also investigate boundary regularity properties.
{"title":"Internal aggregation models with multiple sources and obstacle problems on Sierpiński gaskets","authors":"Uta Freiberg, Nico Heizmann, Robin Kaiser, Ecaterina Sava-Huss","doi":"10.4171/jfg/141","DOIUrl":"https://doi.org/10.4171/jfg/141","url":null,"abstract":"We consider the doubly infinite Sierpiński gasket graph $mathsf{SG}_0$, rescale it by factor $2^{-n}$, and on the rescaled graphs $mathsf{SG}_n=2^{-n}mathsf{SG}0$, for every $ninmathbb{N}$, we investigate the limit shape of three aggregation models with initial configuration $sigma_n$ of particles supported on multiple vertices. The models under consideration are: divisible sandpile in which the excess mass is distributed among the vertices until each vertex is stable and has mass less or equal to one, internal DLA in which particles do random walks until finding an empty site, and rotor aggregation in which particles perform deterministic counterparts of random walks until finding an empty site. We denote by $mathsf{SG}=text{cl}(bigcup{n=0}^{infty}mathsf{SG}_n)$ the infinite Sierpiński gasket, which is a closed subset of $mathbb{R}^2$, for which $mathsf{SG}_n$ represents the level-$n$ approximating graph, and we consider a continuous function $sigmacolon mathsf{SG}tomathbb{N}$. For $sigma$ we solve the obstacle problem, and we describe the noncoincidence set $Dsubset mathsf{SG}$ as the solution of a free boundary problem on the fractal $mathsf{SG}$. If the discrete particle configurations $sigma_n$ on the approximating graphs $mathsf{SG}_n$ converge pointwise to the continuous function $sigma$ on the limit set $mathsf{SG}$, we prove that, as $ntoinfty$, the scaling limits of the three aforementioned models on $mathsf{SG}_n$ starting with initial particle configuration $sigma_n$ converge to the deterministic solution $D$ of the free boundary problem on the limit set $mathsf{SG}subsetmathbb{R}^2$. For $D$ we also investigate boundary regularity properties.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136034325","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 this paper we apply Krein's spectral theory of linear diffusions to study the one-dimensional Liouville Brownian Motion and Liouville Brownian excursions from a given point. As an application we estimate the fractal dimensions of level sets of one-dimensional Liouville Brownian motion as well as various probabilistic asymptotic behaviours of Liouville Brownian motion and Liouville Brownian excursions.
{"title":"Spectral representation of one-dimensional Liouville Brownian Motion and Liouville Brownian excursion","authors":"Xiong Jin","doi":"10.4171/jfg/138","DOIUrl":"https://doi.org/10.4171/jfg/138","url":null,"abstract":"In this paper we apply Krein's spectral theory of linear diffusions to study the one-dimensional Liouville Brownian Motion and Liouville Brownian excursions from a given point. As an application we estimate the fractal dimensions of level sets of one-dimensional Liouville Brownian motion as well as various probabilistic asymptotic behaviours of Liouville Brownian motion and Liouville Brownian excursions.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136079078","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}
This paper seeks conditions that ensure that the attractor of a graph directed iterated function system (GD-IFS) cannot be realised as the attractor of a standard iterated function system (IFS). For a strongly connected directed graph, it is known that, if all directed circuits go through a particular vertex, then for any GD-IFS of similarities on $mathbb{R}$ based on the graph and satisfying the convex open set condition (COSC), its attractor associated with that vertex is also the attractor of a (COSC) standard IFS. In this paper we show the following complementary result. If there exists a directed circuit which does not go through a certain vertex, then there exists a GD-IFS based on the graph such that the attractor associated with that vertex is not the attractor of any standard IFS of similarities. Indeed, we give algebraic conditions for such GD-IFS attractors not to be attractors of standard IFSs, and thus show that `almost-all' COSC GD-IFSs based on the graph have attractors associated with this vertex that are not the attractors of any COSC standard IFS.
{"title":"A dichotomy on the self-similarity of graph-directed attractors","authors":"Kenneth Falconer, Jiaxin Hu, Junda Zhang","doi":"10.4171/jfg/140","DOIUrl":"https://doi.org/10.4171/jfg/140","url":null,"abstract":"This paper seeks conditions that ensure that the attractor of a graph directed iterated function system (GD-IFS) cannot be realised as the attractor of a standard iterated function system (IFS). For a strongly connected directed graph, it is known that, if all directed circuits go through a particular vertex, then for any GD-IFS of similarities on $mathbb{R}$ based on the graph and satisfying the convex open set condition (COSC), its attractor associated with that vertex is also the attractor of a (COSC) standard IFS. In this paper we show the following complementary result. If there exists a directed circuit which does not go through a certain vertex, then there exists a GD-IFS based on the graph such that the attractor associated with that vertex is not the attractor of any standard IFS of similarities. Indeed, we give algebraic conditions for such GD-IFS attractors not to be attractors of standard IFSs, and thus show that `almost-all' COSC GD-IFSs based on the graph have attractors associated with this vertex that are not the attractors of any COSC standard IFS.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136184417","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 introduce the error-sum function of Pierce expansions. Some basic properties of the error-sum function are analyzed. We also examine the fractal property of the graph of it by calculating the Hausdorff dimension, the box-counting dimension, and the covering dimension of the graph.
{"title":"On the error-sum function of Pierce expansions","authors":"Min Woong Ahn","doi":"10.4171/jfg/142","DOIUrl":"https://doi.org/10.4171/jfg/142","url":null,"abstract":"We introduce the error-sum function of Pierce expansions. Some basic properties of the error-sum function are analyzed. We also examine the fractal property of the graph of it by calculating the Hausdorff dimension, the box-counting dimension, and the covering dimension of the graph.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135814256","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 a new and simple method for the determination of the pointwise H"{o}lder exponent of Riemann's function $sum_{n=1}^{infty} n^{-2}sin(pi n^{2} x)$ at every point of the real line. In contrast to earlier approaches, where wavelet analysis and the theta modular group were needed for the analysis of irrational points, our method is direct and elementary, being only based on the following tools from number theory and complex analysis: the evaluation of quadratic Gauss sums, the Poisson summation formula, and Cauchy's theorem.
{"title":"The pointwise behavior of Riemann’s function","authors":"Frederik Broucke, J. Vindas","doi":"10.4171/jfg/137","DOIUrl":"https://doi.org/10.4171/jfg/137","url":null,"abstract":"We present a new and simple method for the determination of the pointwise H\"{o}lder exponent of Riemann's function $sum_{n=1}^{infty} n^{-2}sin(pi n^{2} x)$ at every point of the real line. In contrast to earlier approaches, where wavelet analysis and the theta modular group were needed for the analysis of irrational points, our method is direct and elementary, being only based on the following tools from number theory and complex analysis: the evaluation of quadratic Gauss sums, the Poisson summation formula, and Cauchy's theorem.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46661200","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 study linear and nonlinear Schr¨odinger equations defined by fractal measures. Under the assumption that the Laplacian has compact resolvent, we prove that there exists a uniqueness weak solution for a linear Schr¨odinger equation, and then use it to obtain the existence and uniqueness of weak solution of a nonlinear Schr¨odinger equation. We prove that for a class of self-similar measures on R with overlaps, the Schr¨odinger equations can be discretized so that the finite element method can be applied to obtain approximate solutions. We also prove that the numerical solutions converge to the actual solution and obtain the rate of convergence
{"title":"Schrödinger equations defined by a class of self-similar measures","authors":"Sze-Man Ngai, W. Tang","doi":"10.4171/jfg/134","DOIUrl":"https://doi.org/10.4171/jfg/134","url":null,"abstract":". We study linear and nonlinear Schr¨odinger equations defined by fractal measures. Under the assumption that the Laplacian has compact resolvent, we prove that there exists a uniqueness weak solution for a linear Schr¨odinger equation, and then use it to obtain the existence and uniqueness of weak solution of a nonlinear Schr¨odinger equation. We prove that for a class of self-similar measures on R with overlaps, the Schr¨odinger equations can be discretized so that the finite element method can be applied to obtain approximate solutions. We also prove that the numerical solutions converge to the actual solution and obtain the rate of convergence","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46307419","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}
{"title":"$h$-Laplacians on singular sets","authors":"Claire David, G. Lebeau","doi":"10.4171/jfg/126","DOIUrl":"https://doi.org/10.4171/jfg/126","url":null,"abstract":"","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47171616","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}
{"title":"Fourier decay behavior of homogeneous self-similar measures on the complex plane","authors":"Carolina A. Mosquera, Andrea Olivo","doi":"10.4171/jfg/125","DOIUrl":"https://doi.org/10.4171/jfg/125","url":null,"abstract":"","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46606031","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}
SONG-IL Ri, V. Drakopoulos, Song-Min Nam, Kyong-Mi Kim
{"title":"Nonlinear fractal interpolation functions on the Koch curve","authors":"SONG-IL Ri, V. Drakopoulos, Song-Min Nam, Kyong-Mi Kim","doi":"10.4171/jfg/123","DOIUrl":"https://doi.org/10.4171/jfg/123","url":null,"abstract":"","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49350048","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}
{"title":"Oscillations of BV measures on unbounded nested fractals","authors":"Patricia Alonso Ruiz, Fabrice Baudoin","doi":"10.4171/jfg/122","DOIUrl":"https://doi.org/10.4171/jfg/122","url":null,"abstract":"","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43567946","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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