In this paper we partially answer a question of P. Tukia about the size of the difference between the big horospheric limit set and the horospheric limit set of a Kleinian group. We mainly investigate the case of normal subgroups of Kleinian groups of divergence type and show that this difference is of zero conformal measure by using another result obtained here: the Myrberg limit set of a non-elementary Kleinian group is contained in the horospheric limit set of any non-trivial normal subgroup.
{"title":"On horospheric limit sets of Kleinian groups","authors":"K. Falk, Katsuhiko Matsuzaki","doi":"10.4171/jfg/93","DOIUrl":"https://doi.org/10.4171/jfg/93","url":null,"abstract":"In this paper we partially answer a question of P. Tukia about the size of the difference between the big horospheric limit set and the horospheric limit set of a Kleinian group. We mainly investigate the case of normal subgroups of Kleinian groups of divergence type and show that this difference is of zero conformal measure by using another result obtained here: the Myrberg limit set of a non-elementary Kleinian group is contained in the horospheric limit set of any non-trivial normal subgroup.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47635508","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 construct a subset of $mathbb{R}^d$ which asymptotically and omnidirectionally contains arithmetic progressions but has Assouad dimension 1. More precisely, we say that $F$ asymptotically and omnidirectionally contains arithmetic progressions if we can find an arithmetic progression of length $k$ and gap length $Delta>0$ with direction $ein S^{d-1}$ inside the $epsilon Delta$ neighbourhood of $F$ for all $epsilon>0$, $kgeq 3$ and $ein S^{d-1}$. Moreover, the dimension of our constructed example is the lowest-possible because we prove that a subset of $mathbb{R}^d$ which asymptotically and omnidirectionally contains arithmetic progressions must have Assouad dimension greater than or equal to 1. We also get the same results for arithmetic patches, which are the higher dimensional extension of arithmetic progressions.
{"title":"Construction of a one-dimensional set which asymptotically and omnidirectionally contains arithmetic progressions","authors":"Kota Saito","doi":"10.4171/jfg/90","DOIUrl":"https://doi.org/10.4171/jfg/90","url":null,"abstract":"In this paper, we construct a subset of $mathbb{R}^d$ which asymptotically and omnidirectionally contains arithmetic progressions but has Assouad dimension 1. More precisely, we say that $F$ asymptotically and omnidirectionally contains arithmetic progressions if we can find an arithmetic progression of length $k$ and gap length $Delta>0$ with direction $ein S^{d-1}$ inside the $epsilon Delta$ neighbourhood of $F$ for all $epsilon>0$, $kgeq 3$ and $ein S^{d-1}$. Moreover, the dimension of our constructed example is the lowest-possible because we prove that a subset of $mathbb{R}^d$ which asymptotically and omnidirectionally contains arithmetic progressions must have Assouad dimension greater than or equal to 1. We also get the same results for arithmetic patches, which are the higher dimensional extension of arithmetic progressions.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43763820","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 consider Marstrand type projection theorems for closest-point projections in the normed space $mathbb{R}^2$. We prove that if a norm on $mathbb{R}^2$ is regular enough, then the analogues of the well-known statements from the Euclidean setting hold, while they fail for norms whose unit balls have corners. We establish our results by verifying Peres and Schlag's transversality property and thereby also obtain a Besicovitch-Federer type characterization of purely unrectifiable sets.
{"title":"Marstrand type projection theorems for normed spaces","authors":"Z. Balogh, Annina Iseli","doi":"10.4171/jfg/81","DOIUrl":"https://doi.org/10.4171/jfg/81","url":null,"abstract":"We consider Marstrand type projection theorems for closest-point projections in the normed space $mathbb{R}^2$. We prove that if a norm on $mathbb{R}^2$ is regular enough, then the analogues of the well-known statements from the Euclidean setting hold, while they fail for norms whose unit balls have corners. We establish our results by verifying Peres and Schlag's transversality property and thereby also obtain a Besicovitch-Federer type characterization of purely unrectifiable sets.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/jfg/81","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48681271","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}
Labyrinth fractals are dendrites in the unit square. They were introduced and studied in the last decade first in the self-similar case [Cristea & Steinsky (2009,2011)], then in the mixed case [Cristea & Steinsky (2017), Cristea & Leobacher (2017)]. Supermixed fractals constitute a significant generalisation of mixed labyrinth fractals: each step of the iterative construction is done according to not just one labyrinth pattern, but possibly to several different patterns. In this paper we introduce and study supermixed labyrinth fractals and the corresponding prefractals, called supermixed labyrinth sets, with focus on the aspects that were previously studied for the self-similar and mixed case: topological properties and properties of the arcs between points in the fractal. The facts and formulae found here extend results proven in the above mentioned cases. One of the main results is a sufficient condition for infinite length of arcs in mixed labyrinth fractals.
{"title":"Supermixed labyrinth fractals","authors":"L. Cristea, G. Leobacher","doi":"10.4171/jfg/88","DOIUrl":"https://doi.org/10.4171/jfg/88","url":null,"abstract":"Labyrinth fractals are dendrites in the unit square. They were introduced and studied in the last decade first in the self-similar case [Cristea & Steinsky (2009,2011)], then in the mixed case [Cristea & Steinsky (2017), Cristea & Leobacher (2017)]. Supermixed fractals constitute a significant generalisation of mixed labyrinth fractals: each step of the iterative construction is done according to not just one labyrinth pattern, but possibly to several different patterns. In this paper we introduce and study supermixed labyrinth fractals and the corresponding prefractals, called supermixed labyrinth sets, with focus on the aspects that were previously studied for the self-similar and mixed case: topological properties and properties of the arcs between points in the fractal. The facts and formulae found here extend results proven in the above mentioned cases. One of the main results is a sufficient condition for infinite length of arcs in mixed labyrinth fractals.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43102747","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 consider the self-affine tiles with collinear digit set defined as follows. Let $A,Binmathbb{Z}$ satisfy $|A|leq Bgeq 2$ and $Minmathbb{Z}^{2times2}$ be an integral matrix with characteristic polynomial $x^2+Ax+B$. Moreover, let $mathcal{D}={0,v,2v,ldots,(B-1)v}$ for some $vinmathbb{Z}^2$ such that $v,M v$ are linearly independent. We are interested in the topological properties of the self-affine tile $mathcal{T}$ defined by $Mmathcal{T}=bigcup_{dinmathcal{D}}(mathcal{T}+d)$. Lau and Leung proved that $mathcal{T}$ is homeomorphic to a closed disk if and only if $2|A|leq B+2$. In particular, $mathcal{T}$ has no cut point. We prove here that $mathcal{T}$ has a cut point if and only if $2|A|geq B+5$. For $2|A|-Bin {3,4}$, the interior of $mathcal{T}$ is disconnected and the closure of each connected component of the interior of $mathcal{T}$ is homeomorphic to a closed disk.
{"title":"Topology of planar self-affine tiles with collinear digit set","authors":"S. Akiyama, B. Loridant, J. Thuswaldner","doi":"10.4171/jfg/98","DOIUrl":"https://doi.org/10.4171/jfg/98","url":null,"abstract":"We consider the self-affine tiles with collinear digit set defined as follows. Let $A,Binmathbb{Z}$ satisfy $|A|leq Bgeq 2$ and $Minmathbb{Z}^{2times2}$ be an integral matrix with characteristic polynomial $x^2+Ax+B$. Moreover, let $mathcal{D}={0,v,2v,ldots,(B-1)v}$ for some $vinmathbb{Z}^2$ such that $v,M v$ are linearly independent. We are interested in the topological properties of the self-affine tile $mathcal{T}$ defined by $Mmathcal{T}=bigcup_{dinmathcal{D}}(mathcal{T}+d)$. Lau and Leung proved that $mathcal{T}$ is homeomorphic to a closed disk if and only if $2|A|leq B+2$. In particular, $mathcal{T}$ has no cut point. We prove here that $mathcal{T}$ has a cut point if and only if $2|A|geq B+5$. For $2|A|-Bin {3,4}$, the interior of $mathcal{T}$ is disconnected and the closure of each connected component of the interior of $mathcal{T}$ is homeomorphic to a closed disk.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70871594","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 a formulation of Burgers equation on the Sierpinski gasket, which is the prototype of a p.c.f. self-similar fractal. One possibility is to implement Burgers equation as a semilinear heat equation associated with the Laplacian for scalar functions, just as on the unit interval. Here we propose a second, different formulation which follows from the Cole-Hopf transform and is associated with the Laplacian for vector fields. The difference between these two equations can be understood in terms of different vertex conditions for Laplacians on metric graphs. For the second formulation we show existence and uniqueness of solutions and verify the continuous dependence on the initial condition. We also prove that solutions on the Sierpinski gasket can be approximated in a weak sense by solutions to corresponding equations on approximating metric graphs. These results are part of a larger program discussing nonlinear partial differential equations on fractal spaces.
{"title":"On the viscous Burgers equation on metric graphs and fractals","authors":"Michael Hinz, Melissa Meinert","doi":"10.4171/jfg/87","DOIUrl":"https://doi.org/10.4171/jfg/87","url":null,"abstract":"We study a formulation of Burgers equation on the Sierpinski gasket, which is the prototype of a p.c.f. self-similar fractal. One possibility is to implement Burgers equation as a semilinear heat equation associated with the Laplacian for scalar functions, just as on the unit interval. Here we propose a second, different formulation which follows from the Cole-Hopf transform and is associated with the Laplacian for vector fields. The difference between these two equations can be understood in terms of different vertex conditions for Laplacians on metric graphs. For the second formulation we show existence and uniqueness of solutions and verify the continuous dependence on the initial condition. We also prove that solutions on the Sierpinski gasket can be approximated in a weak sense by solutions to corresponding equations on approximating metric graphs. These results are part of a larger program discussing nonlinear partial differential equations on fractal spaces.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/jfg/87","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42779096","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 2012 Lau and Ngai, motivated by the work of Denker and Sato, gave an example of an isotropic Markov chain on the set of finite words over a three letter alphabet, whose Martin boundary is homeomorphic to the Sierpinski gasket. Here, we extend the results of Lau and Ngai to a class of non-isotropic Markov chains. We determine the Martin boundary and show that the minimal Martin boundary is a proper subset of the Martin boundary. In addition, we give a description of the set of harmonic functions.
{"title":"The Sierpiński gasket as the Martin boundary of a non-isotropic Markov chain","authors":"Marc Kessebohmer, Tony Samuel, K. Sender","doi":"10.4171/JFG/86","DOIUrl":"https://doi.org/10.4171/JFG/86","url":null,"abstract":"In 2012 Lau and Ngai, motivated by the work of Denker and Sato, gave an example of an isotropic Markov chain on the set of finite words over a three letter alphabet, whose Martin boundary is homeomorphic to the Sierpinski gasket. Here, we extend the results of Lau and Ngai to a class of non-isotropic Markov chains. We determine the Martin boundary and show that the minimal Martin boundary is a proper subset of the Martin boundary. In addition, we give a description of the set of harmonic functions.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45197861","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 Bourgain's discretized projection theorem to higher rank situations. Like Bourgain's theorem, our result yields an estimate for the Hausdorff dimension of the exceptional sets in projection theorems formulated in terms of Hausdorff dimensions. This estimate complements earlier results of Mattila and Falconer.
{"title":"Orthogonal projections of discretized sets","authors":"Weikun He","doi":"10.4171/jfg/92","DOIUrl":"https://doi.org/10.4171/jfg/92","url":null,"abstract":"We generalize Bourgain's discretized projection theorem to higher rank situations. Like Bourgain's theorem, our result yields an estimate for the Hausdorff dimension of the exceptional sets in projection theorems formulated in terms of Hausdorff dimensions. This estimate complements earlier results of Mattila and Falconer.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48891338","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 construct a one-parameter family of Laplacians on the Sierpinski Gasket that are symmetric and self-similar for the 9-map iterated function system obtained by iterating the standard 3-map iterated function system. Our main result is the fact that all these Laplacians satisfy a version of spectral decimation that builds a precise catalog of eigenvalues and eigenfunctions for any choice of the parameter. We give a number of applications of this spectral decimation. We also prove analogous results for fractal Laplacians on the unit Interval, and this yields an analogue of the classical Sturm-Liouville theory for the eigenfunctions of these one-dimensional Laplacians.
{"title":"Spectral decimation for families of self-similar symmetric Laplacians on the Sierpiński gasket","authors":"S. Fang, Dylan A. King, E. Lee, R. Strichartz","doi":"10.4171/jfg/83","DOIUrl":"https://doi.org/10.4171/jfg/83","url":null,"abstract":"We construct a one-parameter family of Laplacians on the Sierpinski Gasket that are symmetric and self-similar for the 9-map iterated function system obtained by iterating the standard 3-map iterated function system. Our main result is the fact that all these Laplacians satisfy a version of spectral decimation that builds a precise catalog of eigenvalues and eigenfunctions for any choice of the parameter. We give a number of applications of this spectral decimation. We also prove analogous results for fractal Laplacians on the unit Interval, and this yields an analogue of the classical Sturm-Liouville theory for the eigenfunctions of these one-dimensional Laplacians.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/jfg/83","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43982733","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}
Fractal geometry is the study of sets which exhibit the same pattern at multiple scales. Developing tools to study these sets is of great interest. One step towards developing some of these tools is recognizing the duality between topological spaces and commutative $C^ast$-algebras. When one lifts the commutativity axiom, one gets what are called noncommutative spaces and the study of noncommutative geometry. The tools built to study noncommutative spaces can in fact be used to study fractal sets. In what follows we will use the spectral triples of noncommutative geometry to describe various notions from fractal geometry. We focus on the fractal sets known as the harmonic Sierpinski gasket and the stretched Sierpinski gasket, and show that the spectral triples constructed by Christensen, Ivan, and Lapidus in 2008 and Lapidus and Sarhad in 2015, can recover the standard self-affine measure in the case of the harmonic Sierpinski gasket and the Hausdorff dimension, geodesic metric, and Hausdorff measure in the case of the stretched Sierpinski gasket.
{"title":"Spectral triples for the variants of the Sierpiński gasket","authors":"Andrea Arauza Rivera","doi":"10.4171/JFG/75","DOIUrl":"https://doi.org/10.4171/JFG/75","url":null,"abstract":"Fractal geometry is the study of sets which exhibit the same pattern at multiple scales. Developing tools to study these sets is of great interest. One step towards developing some of these tools is recognizing the duality between topological spaces and commutative $C^ast$-algebras. When one lifts the commutativity axiom, one gets what are called noncommutative spaces and the study of noncommutative geometry. The tools built to study noncommutative spaces can in fact be used to study fractal sets. In what follows we will use the spectral triples of noncommutative geometry to describe various notions from fractal geometry. We focus on the fractal sets known as the harmonic Sierpinski gasket and the stretched Sierpinski gasket, and show that the spectral triples constructed by Christensen, Ivan, and Lapidus in 2008 and Lapidus and Sarhad in 2015, can recover the standard self-affine measure in the case of the harmonic Sierpinski gasket and the Hausdorff dimension, geodesic metric, and Hausdorff measure in the case of the stretched Sierpinski gasket.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44089528","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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