Homogeneous random fractals form a probabilistic extension of self-similar sets with more dependencies than in random recursive constructions. For such random fractals we consider mean values of the Lipschitz-Killing curvatures of their parallel sets for small parallel radii. Under the Uniform Strong Open Set Condition and some further geometric assumptions we show that rescaled limits of these mean values exist as the parallel radius tends to zero. Moreover, integral representations are derived for these limits which extend those known in the deterministic case.
{"title":"Mean Lipschitz--Killing curvatures for homogeneous random fractals","authors":"J. Rataj, S. Winter, M. Zahle","doi":"10.4171/jfg/124","DOIUrl":"https://doi.org/10.4171/jfg/124","url":null,"abstract":"Homogeneous random fractals form a probabilistic extension of self-similar sets with more dependencies than in random recursive constructions. For such random fractals we consider mean values of the Lipschitz-Killing curvatures of their parallel sets for small parallel radii. Under the Uniform Strong Open Set Condition and some further geometric assumptions we show that rescaled limits of these mean values exist as the parallel radius tends to zero. Moreover, integral representations are derived for these limits which extend those known in the deterministic case.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45225350","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 note we introduce a notion of a morphism between two hyperbolic iterated function systems. We prove that the graph of a morphism is the attractor of an iterated function system, giving a Closed Graph Theorem, and show how it can be used to approach the topological conjugacy problem for iterated function systems.
{"title":"A closed graph theorem for hyperbolic iterated function systems","authors":"A. Mundey","doi":"10.4171/JFG/116","DOIUrl":"https://doi.org/10.4171/JFG/116","url":null,"abstract":"In this note we introduce a notion of a morphism between two hyperbolic iterated function systems. We prove that the graph of a morphism is the attractor of an iterated function system, giving a Closed Graph Theorem, and show how it can be used to approach the topological conjugacy problem for iterated function systems.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47538485","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}
Let $M$ be a positive integer and $qin (1, M+1]$. A $q$-expansion of a real number $x$ is a sequence $(c_i)=c_1c_2cdots$ with $c_iin {0,1,ldots, M}$ such that $x=sum_{i=1}^{infty}c_iq^{-i}$. In this paper we study the set $mathcal{U}_q^j$ consisting of those real numbers having exactly $j$ $q$-expansions. Our main result is that for Lebesgue almost every $qin (q_{KL}, M+1), $ we have $$dim_{H}mathcal{U}_{q}^{j}leq max{0, 2dim_Hmathcal{U}_q-1}text{ for all } jin{2,3,ldots}.$$ Here $q_{KL}$ is the Komornik-Loreti constant. As a corollary of this result, we show that for any $jin{2,3,ldots},$ the function mapping $q$ to $dim_{H}mathcal{U}_{q}^{j}$ is not continuous.
设$M$为正整数，$qin (1, M+1]$。实数$x$的$q$ -展开是一个含有$c_iin {0,1,ldots, M}$的序列$(c_i)=c_1c_2cdots$，使得$x=sum_{i=1}^{infty}c_iq^{-i}$。本文研究了由恰好具有$j$$q$ -展开式的实数组成的集合$mathcal{U}_q^j$。我们的主要结果是，对于勒贝格，几乎每一个$qin (q_{KL}, M+1), $我们都有$$dim_{H}mathcal{U}_{q}^{j}leq max{0, 2dim_Hmathcal{U}_q-1}text{ for all } jin{2,3,ldots}.$$这里$q_{KL}$是Komornik-Loreti常数。作为这个结果的推论，我们证明对于任何$jin{2,3,ldots},$，将$q$映射到$dim_{H}mathcal{U}_{q}^{j}$的函数是不连续的。
{"title":"Metric results for numbers with multiple $q$-expansions","authors":"S. Baker, Yuru Zou","doi":"10.4171/jfg/131","DOIUrl":"https://doi.org/10.4171/jfg/131","url":null,"abstract":"Let $M$ be a positive integer and $qin (1, M+1]$. A $q$-expansion of a real number $x$ is a sequence $(c_i)=c_1c_2cdots$ with $c_iin {0,1,ldots, M}$ such that $x=sum_{i=1}^{infty}c_iq^{-i}$. In this paper we study the set $mathcal{U}_q^j$ consisting of those real numbers having exactly $j$ $q$-expansions. Our main result is that for Lebesgue almost every $qin (q_{KL}, M+1), $ we have $$dim_{H}mathcal{U}_{q}^{j}leq max{0, 2dim_Hmathcal{U}_q-1}text{ for all } jin{2,3,ldots}.$$ Here $q_{KL}$ is the Komornik-Loreti constant. As a corollary of this result, we show that for any $jin{2,3,ldots},$ the function mapping $q$ to $dim_{H}mathcal{U}_{q}^{j}$ is not continuous.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70871381","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}
Given a non-empty bounded subset of hyperbolic space and a Kleinian group acting on that space, the orbital set is the orbit of the given set under the action of the group. We may view orbital sets as bounded (often fractal) subsets of Euclidean space. We prove that the upper box dimension of an orbital set is given by the maximum of three quantities: the upper box dimension of the given set; the Poincar'e exponent of the Kleinian group; and the upper box dimension of the limit set of the Kleinian group. Since we do not make any assumptions about the Kleinian group, none of the terms in the maximum can be removed in general. We show by constructing an explicit example that the (hyperbolic) boundedness assumption on $C$ cannot be removed in general.
{"title":"Dimensions of Kleinian orbital sets","authors":"T. Bartlett, J. Fraser","doi":"10.4171/jfg/139","DOIUrl":"https://doi.org/10.4171/jfg/139","url":null,"abstract":"Given a non-empty bounded subset of hyperbolic space and a Kleinian group acting on that space, the orbital set is the orbit of the given set under the action of the group. We may view orbital sets as bounded (often fractal) subsets of Euclidean space. We prove that the upper box dimension of an orbital set is given by the maximum of three quantities: the upper box dimension of the given set; the Poincar'e exponent of the Kleinian group; and the upper box dimension of the limit set of the Kleinian group. Since we do not make any assumptions about the Kleinian group, none of the terms in the maximum can be removed in general. We show by constructing an explicit example that the (hyperbolic) boundedness assumption on $C$ cannot be removed in general.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45542407","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}
lim p μpξq “ 0, as |ξ| Ñ 8, where |ξ| is a norm (say, the Euclidean norm) of ξ P Rd. Whereas absolutely continuous measures are Rajchman by the Riemann-Lebesgue Lemma, it is a subtle question to decide which singular measures are such, see, e.g., the survey of Lyons [14]. A much stronger property, useful for many applications is the following. Definition 1.1. For α ą 0 let Ddpαq “ ν finite positive measure on Rd : |p νptq| “ Oνp|t|q, |t| Ñ 8 (
{"title":"Fourier decay for homogeneous self-affine measures","authors":"B. Solomyak","doi":"10.4171/jfg/119","DOIUrl":"https://doi.org/10.4171/jfg/119","url":null,"abstract":"lim p μpξq “ 0, as |ξ| Ñ 8, where |ξ| is a norm (say, the Euclidean norm) of ξ P Rd. Whereas absolutely continuous measures are Rajchman by the Riemann-Lebesgue Lemma, it is a subtle question to decide which singular measures are such, see, e.g., the survey of Lyons [14]. A much stronger property, useful for many applications is the following. Definition 1.1. For α ą 0 let Ddpαq “ ν finite positive measure on Rd : |p νptq| “ Oνp|t|q, |t| Ñ 8 (","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49144934","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 multipliers and transfer operators","authors":"M. Pollicott","doi":"10.4171/JFG/103","DOIUrl":"https://doi.org/10.4171/JFG/103","url":null,"abstract":"","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45896725","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 the asymptotic solution of the equation of the pressure function $smapsto P(svarphi(epsilon,cdot)+psi(epsilon,cdot))$ for perturbed potentials $varphi(epsilon,cdot)$ and $psi(epsilon,cdot)$ defined on the shift space with countable state space. In our main result, we give a sufficient condition for the solution $s=s(epsilon)$ of $P(svarphi(epsilon,cdot)+psi(epsilon,cdot))=0$ to have the $n$-order asymptotic expansion for the small parameter $epsilon$. In addition, we also obtain the case where the order of the expansion of the solution $s=s(epsilon)$ is less than the order of the expansion of the perturbed potentials. Our results can be applied to problems concerning asymptotic behaviors of Hausdorff dimensions obtained from Bowen formula: conformal graph directed Markov systems, an infinite graph directed systems with contractive infinitesimal similitudes mappings, and other concrete examples.
{"title":"Asymptotic solution of Bowen equation for perturbed potentials on shift spaces with countable states","authors":"Haruyoshi Tanaka","doi":"10.4171/jfg/128","DOIUrl":"https://doi.org/10.4171/jfg/128","url":null,"abstract":"We study the asymptotic solution of the equation of the pressure function $smapsto P(svarphi(epsilon,cdot)+psi(epsilon,cdot))$ for perturbed potentials $varphi(epsilon,cdot)$ and $psi(epsilon,cdot)$ defined on the shift space with countable state space. In our main result, we give a sufficient condition for the solution $s=s(epsilon)$ of $P(svarphi(epsilon,cdot)+psi(epsilon,cdot))=0$ to have the $n$-order asymptotic expansion for the small parameter $epsilon$. In addition, we also obtain the case where the order of the expansion of the solution $s=s(epsilon)$ is less than the order of the expansion of the perturbed potentials. Our results can be applied to problems concerning asymptotic behaviors of Hausdorff dimensions obtained from Bowen formula: conformal graph directed Markov systems, an infinite graph directed systems with contractive infinitesimal similitudes mappings, and other concrete examples.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48940953","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 a general ampliation homogeneity result for the quasicentral modulus of an n-tuple of operators with respect to the (p,1) Lorentz normed ideal. We use this to prove a formula involving Hausdorff measure for the quasicentral modulus of n-tuples of commuting Hermitian operators the spectrum of which is contained in certain Cantor-like self-similar fractals.
{"title":"The formula for the quasicentral modulus in the case of spectral measures on fractals","authors":"D. Voiculescu","doi":"10.4171/jfg/108","DOIUrl":"https://doi.org/10.4171/jfg/108","url":null,"abstract":"We prove a general ampliation homogeneity result for the quasicentral modulus of an n-tuple of operators with respect to the (p,1) Lorentz normed ideal. We use this to prove a formula involving Hausdorff measure for the quasicentral modulus of n-tuples of commuting Hermitian operators the spectrum of which is contained in certain Cantor-like self-similar fractals.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42439920","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 conditions on a general family $P_{lambda}:R^ntoR^m, lambda in Lambda,$ of orthogonal projections which guarantee that the Hausdorff dimension formula $dim Acap P_{lambda}^{-1}{u}=s-m$ holds generically for measurable sets $AsubsetRn$ with positive and finite $s$-dimensional Hausdorff measure, $s>m$, and with positive lower density. As an application we prove for measurable sets $A,BsubsetRn$ with positive $s$- and $t$-dimensional measures, and with positive lower density that if $s + (n-1)t/n > n$, then $dim Acap (g(B)+z) = s+t - n$ for almost all rotations $g$ and for positively many $zinRn$.
{"title":"Hausdorff dimension of intersections with planes and general sets","authors":"P. Mattila","doi":"10.4171/jfg/110","DOIUrl":"https://doi.org/10.4171/jfg/110","url":null,"abstract":"We give conditions on a general family $P_{lambda}:R^ntoR^m, lambda in Lambda,$ of orthogonal projections which guarantee that the Hausdorff dimension formula $dim Acap P_{lambda}^{-1}{u}=s-m$ holds generically for measurable sets $AsubsetRn$ with positive and finite $s$-dimensional Hausdorff measure, $s>m$, and with positive lower density. As an application we prove for measurable sets $A,BsubsetRn$ with positive $s$- and $t$-dimensional measures, and with positive lower density that if $s + (n-1)t/n > n$, then $dim Acap (g(B)+z) = s+t - n$ for almost all rotations $g$ and for positively many $zinRn$.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47486320","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}
Terence L. J. Harris, Chi N. Y. Huynh, Fernando Roman-Garcia
We study the family of vertical projections whose fibers are right cosets of horizontal planes in the Heisenberg group, $mathbb{H}^n$. We prove lower bounds for Hausdorff dimension distortion of sets under these mappings, with respect to the Euclidean metric and also the natural quotient metric. We show these bounds are sharp in a large part of the range of possible dimension, and give conjectured sharp lower bounds for the remaining part of the range. Our result also lets us improve the known almost sure lower bound for the standard family of vertical projections in $mathbb{H}^n$.
{"title":"Dimension distortion by right coset projections in the Heisenberg group","authors":"Terence L. J. Harris, Chi N. Y. Huynh, Fernando Roman-Garcia","doi":"10.4171/JFG/106","DOIUrl":"https://doi.org/10.4171/JFG/106","url":null,"abstract":"We study the family of vertical projections whose fibers are right cosets of horizontal planes in the Heisenberg group, $mathbb{H}^n$. We prove lower bounds for Hausdorff dimension distortion of sets under these mappings, with respect to the Euclidean metric and also the natural quotient metric. We show these bounds are sharp in a large part of the range of possible dimension, and give conjectured sharp lower bounds for the remaining part of the range. Our result also lets us improve the known almost sure lower bound for the standard family of vertical projections in $mathbb{H}^n$.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41884614","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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