We study equilibrium measures (Kaenmaki measures) supported on self-affine sets generated by a finite collection of diagonal and anti-diagonal matrices acting on the plane and satisfying the strong separation property. Our main result is that such measures are exact dimensional and the dimension satisfies the Ledrappier-Young formula, which gives an explicit expression for the dimension in terms of the entropy and Lyapunov exponents as well as the dimension of the important coordinate projection of the measure. In particular, we do this by showing that the Kaenmaki measure is equal to the sum of (the pushforwards) of two Gibbs measures on an associated subshift of finite type.
{"title":"Dimensions of equilibrium measures on a class of planar self-affine sets","authors":"J. Fraser, T. Jordan, Natalia Jurga","doi":"10.4171/JFG/85","DOIUrl":"https://doi.org/10.4171/JFG/85","url":null,"abstract":"We study equilibrium measures (Kaenmaki measures) supported on self-affine sets generated by a finite collection of diagonal and anti-diagonal matrices acting on the plane and satisfying the strong separation property. Our main result is that such measures are exact dimensional and the dimension satisfies the Ledrappier-Young formula, which gives an explicit expression for the dimension in terms of the entropy and Lyapunov exponents as well as the dimension of the important coordinate projection of the measure. In particular, we do this by showing that the Kaenmaki measure is equal to the sum of (the pushforwards) of two Gibbs measures on an associated subshift of finite type.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41584393","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 positive integer $M$ and $qin(1,M+1]$, let $mathcal U_q$ be the set of $xin[0, M/(q-1)]$ having a unique $q$-expansion: there exists a unique sequence $(x_i)=x_1x_2ldots$ with each $x_iin{0,1,ldots, M}$ such that [ x=frac{x_1}{q}+frac{x_2}{q^2}+frac{x_3}{q^3}+cdots. ] Denote by $mathbf U_q$ the set of corresponding sequences of all points in $mathcal U_q$. It is well-known that the function $H: qmapsto h(mathbf U_q)$ is a Devil's staircase, where $h(mathbf U_q)$ denotes the topological entropy of $mathbf U_q$. In this paper we {give several characterizations of} the bifurcation set [ mathcal B:={qin(1,M+1]: H(p)ne H(q)textrm{ for any }pne q}. ] Note that $mathcal B$ is contained in the set $mathcal{U}^R$ of bases $qin(1,M+1]$ such that $1inmathcal U_q$. By using a transversality technique we also calculate the Hausdorff dimension of the difference $mathcal Bbackslashmathcal{U}^R$. Interestingly this quantity is always strictly between $0$ and $1$. When $M=1$ the Hausdorff dimension of $mathcal Bbackslashmathcal{U}^R$ is $frac{log 2}{3log lambda^*}approx 0.368699$, where $lambda^*$ is the unique root in $(1, 2)$ of the equation $x^5-x^4-x^3-2x^2+x+1=0$.
{"title":"Bifurcation sets arising from non-integer base expansions","authors":"P. Allaart, S. Baker, D. Kong","doi":"10.4171/jfg/79","DOIUrl":"https://doi.org/10.4171/jfg/79","url":null,"abstract":"Given a positive integer $M$ and $qin(1,M+1]$, let $mathcal U_q$ be the set of $xin[0, M/(q-1)]$ having a unique $q$-expansion: there exists a unique sequence $(x_i)=x_1x_2ldots$ with each $x_iin{0,1,ldots, M}$ such that \u0000[ \u0000x=frac{x_1}{q}+frac{x_2}{q^2}+frac{x_3}{q^3}+cdots. \u0000] \u0000Denote by $mathbf U_q$ the set of corresponding sequences of all points in $mathcal U_q$. \u0000It is well-known that the function $H: qmapsto h(mathbf U_q)$ is a Devil's staircase, where $h(mathbf U_q)$ denotes the topological entropy of $mathbf U_q$. In this paper we {give several characterizations of} the bifurcation set \u0000[ \u0000mathcal B:={qin(1,M+1]: H(p)ne H(q)textrm{ for any }pne q}. \u0000] Note that $mathcal B$ is contained in the set $mathcal{U}^R$ of bases $qin(1,M+1]$ such that $1inmathcal U_q$. By using a transversality technique we also calculate the Hausdorff dimension of the difference $mathcal Bbackslashmathcal{U}^R$. Interestingly this quantity is always strictly between $0$ and $1$. When $M=1$ the Hausdorff dimension of $mathcal Bbackslashmathcal{U}^R$ is $frac{log 2}{3log lambda^*}approx 0.368699$, where $lambda^*$ is the unique root in $(1, 2)$ of the equation $x^5-x^4-x^3-2x^2+x+1=0$.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/jfg/79","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46154559","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 article we investigate the pressure function and affinity dimension for iterated function systems associated to the "box-like" self-affine fractals investigated by D.-J. Feng, Y. Wang and J.M. Fraser. Combining previous results of V. Yu. Protasov, A. K"aenm"aki and the author we obtain an explicit formula for the pressure function which makes it straightforward to compute the affinity dimension of box-like self-affine sets. We also prove a variant of this formula which allows the computation of a modified singular value pressure function defined by J.M. Fraser. We give some explicit examples where the Hausdorff and packing dimensions of a box-like self-affine fractal may be easily computed.
{"title":"An explicit formula for the pressure of box-like affine iterated function systems","authors":"I. Morris","doi":"10.4171/JFG/72","DOIUrl":"https://doi.org/10.4171/JFG/72","url":null,"abstract":"In this article we investigate the pressure function and affinity dimension for iterated function systems associated to the \"box-like\" self-affine fractals investigated by D.-J. Feng, Y. Wang and J.M. Fraser. Combining previous results of V. Yu. Protasov, A. K\"aenm\"aki and the author we obtain an explicit formula for the pressure function which makes it straightforward to compute the affinity dimension of box-like self-affine sets. We also prove a variant of this formula which allows the computation of a modified singular value pressure function defined by J.M. Fraser. We give some explicit examples where the Hausdorff and packing dimensions of a box-like self-affine fractal may be easily computed.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JFG/72","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44194155","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 show that a set of small box counting dimension can be covered by a H"older graph from all but a small set of directions, and give sharp bounds for the dimension of the exceptional set, improving a result of B. Hunt and V. Kaloshin. We observe that, as a consequence, H"older graphs can have positive doubling measure, answering a question of T. Ojala and T. Rajala. We also give remarks on H"older coverings in polar coordinates and, on the other hand, prove that a Homogenous set of small box counting dimension can be covered by a Lipschitz graph from all but a small set of directions.
{"title":"Hölder coverings of sets of small dimension","authors":"Eino Rossi, Pablo Shmerkin","doi":"10.4171/JFG/78","DOIUrl":"https://doi.org/10.4171/JFG/78","url":null,"abstract":"We show that a set of small box counting dimension can be covered by a H\"older graph from all but a small set of directions, and give sharp bounds for the dimension of the exceptional set, improving a result of B. Hunt and V. Kaloshin. We observe that, as a consequence, H\"older graphs can have positive doubling measure, answering a question of T. Ojala and T. Rajala. We also give remarks on H\"older coverings in polar coordinates and, on the other hand, prove that a Homogenous set of small box counting dimension can be covered by a Lipschitz graph from all but a small set of directions.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47713686","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}
A general class of finitely ramified fractals is that of P.C.F. self-similar sets. An important open problem in analysis on fractals was whether there exists a self-similar energy on every P.C.F. self-similar set. In this paper, I solve the problem, showing an example of a P.C.F. self-similar set where there exists no self-similar energy.
{"title":"A p.c.f. self-similar set with no self-similar energy","authors":"Roberto Peirone","doi":"10.4171/jfg/82","DOIUrl":"https://doi.org/10.4171/jfg/82","url":null,"abstract":"A general class of finitely ramified fractals is that of P.C.F. self-similar sets. An important open problem in analysis on fractals was whether there exists a self-similar energy on every P.C.F. self-similar set. In this paper, I solve the problem, showing an example of a P.C.F. self-similar set where there exists no self-similar energy.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/jfg/82","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46887839","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}
If $n geq 3$ and $Gamma$ is a convex-cocompact Zariski-dense discrete subgroup of $mathbf{SO}^o(1,n+1)$ such that $delta_Gamma=n-m$ where $m$ is an integer, $1 leq m leq n-1$, we show that for any $m$-dimensional subgroup $U$ in the horospheric group $N$, the Burger-Roblin measure associated to $Gamma$ on the quotient of the frame bundle is $U$-recurrent.
{"title":"The case of equality in the dichotomy of Mohammadi–Oh","authors":"Laurent Dufloux","doi":"10.4171/jfg/80","DOIUrl":"https://doi.org/10.4171/jfg/80","url":null,"abstract":"If $n geq 3$ and $Gamma$ is a convex-cocompact Zariski-dense discrete subgroup of $mathbf{SO}^o(1,n+1)$ such that $delta_Gamma=n-m$ where $m$ is an integer, $1 leq m leq n-1$, we show that for any $m$-dimensional subgroup $U$ in the horospheric group $N$, the Burger-Roblin measure associated to $Gamma$ on the quotient of the frame bundle is $U$-recurrent.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48381269","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":"Hausdorff dimension of unions of affine subspaces and of Furstenberg-type sets","authors":"K. Héra, Tamás Keleti, András Máthé","doi":"10.4171/JFG/77","DOIUrl":"https://doi.org/10.4171/JFG/77","url":null,"abstract":"We prove that for any $1 le k<n$ and $sle 1$, the union of any nonempty $s$-Hausdorff dimensional family of $k$-dimensional affine subspaces of ${mathbb R}^n$ has Hausdorff dimension $k+s$. More generally, we show that for any $0 < alpha le k$, if $B subset {mathbb R}^n$ and $E$ is a nonempty collection of $k$-dimensional affine subspaces of ${mathbb R}^n$ such that every $P in E$ intersects $B$ in a set of Hausdorff dimension at least $alpha$, then $dim B ge 2 alpha - k + min(dim E, 1)$, where $dim$ denotes the Hausdorff dimension. As a consequence, we generalize the well known Furstenberg-type estimate that every $alpha$-Furstenberg set has Hausdorff dimension at least $2 alpha$; we strengthen a theorem of Falconer and Mattila; and we show that for any $0 le k<n$, if a set $A subset {mathbb R}^n$ contains the $k$-skeleton of a rotated unit cube around every point of ${mathbb R}^n$, or if $A$ contains a $k$-dimensional affine subspace at a fixed positive distance from every point of ${mathbb R}^n$, then the Hausdorff dimension of $A$ is at least $k + 1$.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JFG/77","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46219077","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 Sierpinski gasket is known to support an exotic stochastic process called the asymptotically one-dimensional diffusion. This process displays local anisotropy, as there is a preferred direction of motion which dominates at the microscale, but on the macroscale we see global isotropy in that the process will behave like the canonical Brownian motion on the fractal. In this paper we analyse the microscale behaviour of such processes, which we call non-fixed point diffusions, for a class of fractals and show that there is a natural limit diffusion associated with the small scale asymptotics. This limit process no longer lives on the original fractal but is supported by another fractal, which is the Gromov-Hausdorff limit of the original set after a shorting operation is performed on the dominant microscale direction of motion. We establish the weak convergence of the rescaled diffusions in a general setting and use this to answer a question raised in Hattori (1994) about the ultraviolet limit of the asymptotically one-dimensional diffusion process on the Sierpinski gasket.
{"title":"Degenerate limits for one-parameter families of non-fixed-point diffusions on fractals","authors":"B. Hambly, Weiye Yang","doi":"10.4171/JFG/67","DOIUrl":"https://doi.org/10.4171/JFG/67","url":null,"abstract":"The Sierpinski gasket is known to support an exotic stochastic process called the asymptotically one-dimensional diffusion. This process displays local anisotropy, as there is a preferred direction of motion which dominates at the microscale, but on the macroscale we see global isotropy in that the process will behave like the canonical Brownian motion on the fractal. In this paper we analyse the microscale behaviour of such processes, which we call non-fixed point diffusions, for a class of fractals and show that there is a natural limit diffusion associated with the small scale asymptotics. This limit process no longer lives on the original fractal but is supported by another fractal, which is the Gromov-Hausdorff limit of the original set after a shorting operation is performed on the dominant microscale direction of motion. We establish the weak convergence of the rescaled diffusions in a general setting and use this to answer a question raised in Hattori (1994) about the ultraviolet limit of the asymptotically one-dimensional diffusion process on the Sierpinski gasket.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2016-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JFG/67","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70871655","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 determine the walk dimension of the Sierpinski gasket without using diffusion. We construct non-local regular Dirichlet forms on the Sierpinski gasket from regular Dirichlet forms on the Sierpinski graph whose suitable boundary is the Sierpinski gasket.
{"title":"Determination of the walk dimension of the Sierpiński gasket without using diffusion","authors":"A. Grigor’yan, Meng Yang","doi":"10.4171/JFG/66","DOIUrl":"https://doi.org/10.4171/JFG/66","url":null,"abstract":"We determine the walk dimension of the Sierpinski gasket without using diffusion. We construct non-local regular Dirichlet forms on the Sierpinski gasket from regular Dirichlet forms on the Sierpinski graph whose suitable boundary is the Sierpinski gasket.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2016-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JFG/66","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70871879","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 spectral properties of the transfer operators $L$ defined on the circle $mathbb T=mathbb R/mathbb Z$ by $$(Lu)(t)=frac{1}{d}sum_{i=0}^{d-1} fleft(frac{t+i}{d}right)uleft(frac{t+i}{d}right), tinmathbb T$$ where $u$ is a function on $mathbb T$. We focus in particular on the cases $f(t)=|cos(pi t)|^q$ and $f(t)=|sin(pi t)|^q$, which are closely related to some classical Fourier-analytic questions. We also obtain some explicit computations, particularly in the case $d=2$. Our study extends work of Strichartz cite{Strichartz1990} and Fan and Lau cite{FanLau1998}.
我们研究了$$(Lu)(t)=frac{1}{d}sum_{i=0}^{d-1} fleft(frac{t+i}{d}right)uleft(frac{t+i}{d}right), tinmathbb T$$在圆$mathbb T=mathbb R/mathbb Z$上定义的传递算子$L$的谱性质，其中$u$是$mathbb T$上的一个函数。我们特别关注与一些经典傅立叶分析问题密切相关的情况$f(t)=|cos(pi t)|^q$和$f(t)=|sin(pi t)|^q$。我们也得到了一些显式的计算，特别是在$d=2$的情况下。我们的研究扩展了Strichartz cite{Strichartz1990}和Fan and Lau cite{FanLau1998}的工作。
{"title":"On transfer operators on the circle with trigonometric weights","authors":"Xianghong Chen, H. Volkmer","doi":"10.4171/JFG/64","DOIUrl":"https://doi.org/10.4171/JFG/64","url":null,"abstract":"We study spectral properties of the transfer operators $L$ defined on the circle $mathbb T=mathbb R/mathbb Z$ by $$(Lu)(t)=frac{1}{d}sum_{i=0}^{d-1} fleft(frac{t+i}{d}right)uleft(frac{t+i}{d}right), tinmathbb T$$ where $u$ is a function on $mathbb T$. We focus in particular on the cases $f(t)=|cos(pi t)|^q$ and $f(t)=|sin(pi t)|^q$, which are closely related to some classical Fourier-analytic questions. We also obtain some explicit computations, particularly in the case $d=2$. Our study extends work of Strichartz cite{Strichartz1990} and Fan and Lau cite{FanLau1998}.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2016-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JFG/64","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70871817","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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