We present a new method to calculate the Hausdorff dimension of a certain class of fractals: boundaries of self-affine tiles. Among the interesting aspects are that even if the affine contraction underlying the iterated function system is not conjugated to a similarity we obtain an upper- and lower-bounds for its Hausdorff dimension. In fact, we obtain the exact value for the dimension if the moduli of the eigenvalues of the underlying affine contraction are all equal (this includes Jordan blocks). The tiles we discuss play an important role in the theory of wavelets. We calculate the dimension for a number of examples.
We investigate the connection between radix representations for Z^n and self-affine tilings of R^n. We apply our results to show that Haar-like multivariable wavelets exist for all dilation matrices that are sufficie
Let $X=\bigcupφ_{i}X$ be a strongly separated self-affine set in $\mathbb{R}^2$ (or one satisfying the strong open set condition). Under mild non-compactness and irreducibility assumptions on the matrix parts of the $φ_{i}$, we prove that $\dim X$ is equal to the affinity dimension, and similarly for self-affine measures and the Lyapunov dimension. The proof is via analysis of the dimension of the orthogonal projections of the measures, and relies on additive combinatorics methods.
An integral self-affine tile is the solution of a set equation $\mathbf{A} \mathcal{T} = \bigcup_{d \in \mathcal{D}} (\mathcal{T} + d)$, where $\mathbf{A}$ is an $n \times n$ integer matrix and $\mathcal{D}$ is a finite subset of $\mathbb{Z}^n$. In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices $\mathbf{A} \in \mathbb{Q}^{n \times n}$. We define rational self-affine tiles as compact subsets of the open subring $\mathbb{R}^n\times \prod_\mathfrak{p} K_\mathfrak{p}$ of the adéle ring $\mathbb{A}_K$, where the factors …
We prove that if $μ$ is a self-affine measure in the plane whose defining IFS acts totally irreducibly on $\mathbb{RP}^1$ and satisfies an exponential separation condition, then its dimension is equal to its Lyapunov dimension. We also treat a class of reducible systems. This extends our previous work on the subject with Bárány to the overlapping case.
We develop tools to study the topology and geometry of self-affine fractals in dimension three and higher. We use the self-affine structure and obtain rather detailed information about the connectedness of interior and boundary sets, and on the dimensions and intersections of boundary sets. As an application, we describe in algebraic terms the polyhedral structure of the six fractal three-dimensional twindragons. Only two of them can be homeomorphic to a ball but even these have faces which are not homeomorphic to a disk.
Using methods from ergodic theory along with properties of the Furstenberg measure we obtain conditions under which certain classes of plane self-affine sets have Hausdorff or box-counting dimensions equal to their affinity dimension. We exhibit some new specific classes of self-affine sets for which these dimensions are equal.
The Hausdorff dimension of general Sierpinski carpets, [4] and [20], and the generalization on Lalley-Gatzouras carpets, [10], are today well known results, the formulas being obtain via the variational principle for the dimension. We call the multidimensional versions of these carpets Sierpinski sponges and self-affine sponges, respectively,. In this paper we show that the Hausdorff dimension of self-affine sponges, defined in R3, is a Lipschitz continuous function at Sierpinski sponges.
We construct a self-affine sponge in $\mathbb R^3$ whose dynamical dimension, i.e. the supremum of the Hausdorff dimensions of its invariant measures, is strictly less than its Hausdorff dimension. This resolves a long-standing open problem in the dimension theory of dynamical systems, namely whether every expanding repeller has an ergodic invariant measure of full Hausdorff dimension. More generally we compute the Hausdorff and dynamical dimensions of a large class of self-affine sponges, a problem that previous techniques could only solve in two dimensions. The Hausdorff and dynamical dimens…
We consider the self-affine tiles with collinear digit set defined as follows. Let $A,B\in\mathbb{Z}$ satisfy $|A|\leq B\geq 2$ and $M\in\mathbb{Z}^{2\times2}$ be an integral matrix with characteristic polynomial $x^2+Ax+B$. Moreover, let $\mathcal{D}=\{0,v,2v,\ldots,(B-1)v\}$ for some $v\in\mathbb{Z}^2$ such that $v,M v$ are linearly independent. We are interested in the topological properties of the self-affine tile $\mathcal{T}$ defined by $M\mathcal{T}=\bigcup_{d\in\mathcal{D}}(\mathcal{T}+d)$. Lau and Leung proved that $\mathcal{T}$ is homeomorphic to a closed disk if and only if $2|A|\le…
We compute the Hausdorff dimension of limit sets generated by 3-dimensional self-affine mappings with diagonal matrices of the form A_{ijk}=Diag(a_{ijk}, b_{ij}, c_{i}), where 0<a_{ijk}\le b_{ij}\le c_i<1. By doing so we show that the variational principle for the dimension holds for this class.
In this work we are interested in the self--affine fractals studied by Gatzouras and Lalley and by the author which generalize the famous general Sierpinski carpets studied by Bedford and McMullen. We give a formula for the Hausdorff dimension of sets which are randomly generated using a finite number of self-affine transformations each one generating a fractal set as mentioned before, with some technical hypotheses. The choice of the transformation is random according to a Bernoulli measure. The formula is given in terms of the variational principle for the dimension.
Let $A$ be an $n\times n$ real expanding matrix and $\mathcal{D}$ be a finite subset of $\mathbb{R}^n$ with $0\in\mathcal{D}$. The family of maps $\{f_d(x)=A^{-1}(x+d)\}_{d\in\mathcal{D}}$ is called a self-affine iterated function system (self-affine IFS). The self-affine set $K=K(A,\mathcal{D})$ is the unique compact set determined by $(A, {\mathcal D})$ satisfying the set-valued equation $K=\displaystyle\bigcup_{d\in\mathcal{D}}f_d(K)$. The number $s=n\,\ln(\# \mathcal{D})/\ln(q)$ with $q=|\det(A)|$, is the so-called pseudo similarity dimension of $K$. As shown by He and Lau, one can associa…
We study two-digit attractors (2-attractors) in $\mathbb{R}^d$ which are self-affine compact sets defined by two contraction affine mappings with the same linear part. They are widely studied in the literature under various names: twindragons, two-digit tiles, 2-reptiles, etc., due to many applications in approximation theory, in the construction of multivariate Haar systems and other wavelet bases, in the discrete geometry, and in the number theory. We obtain a complete classification of isotropic 2-attractors in $\mathbb{R}^d$ and show that they are all homeomorphic but not diffeomorphic. In…
We study the orthogonal projections of a large class of self-affine carpets, which contains the carpets of Bedford and McMullen as special cases. Our main result is that if $Λ$ is such a carpet, and certain natural irrationality conditions hold, then every orthogonal projection of $Λ$ in a non-principal direction has Hausdorff dimension $\min(γ,1)$, where $γ$ is the Hausdorff dimension of $Λ$. This generalizes a recent result of Peres and Shmerkin on sums of Cantor sets.
We compute the Hausdorff dimension of any random statistically self-affine Sierpinski sponge $K\subset \mathbb{R}^k$ ($k\ge 2$) obtained by using some percolation process in $[0,1]^k$. To do so, we first exhibit a Ledrappier-Young type formula for the Hausdorff dimensions of statistically self-affine measures supported on $K$. This formula presents a new feature compared to its deterministic or random dynamical version. Then, we establish a variational principle expressing $\dim_H K$ as the supremum of the Hausdorff dimensions of statistically self-affine measures supported on $K$, and show th…
Let $\textbf{a}_1,\dots, \textbf{a}_r$ be vectors in a half-space of $\mathbb{R}^n$. We call $$C=\textbf{a}_1\mathbb{R}^++\cdots+\textbf{a}_r \mathbb{R}^+$$ a convex polyhedral cone, and call $\{\textbf{a}_1,\dots, \textbf{a}_r\}$ a generator set of $C$. A generator set with the minimal cardinality is called a frame. We investigate the translation tilings of convex polyhedral cones. Let $T\subset \mathbb{R}^n$ be a compact set such that $T$ is the closure of its interior, and $\mathcal{J}\subset \mathbb{R}^n$ be a discrete set. We say $(T,\mathcal{J})$ is a translation tiling of $C$ if $T+\mat…
We prove that for random affine code tree fractals the affinity dimension is almost surely equal to the unique zero of the pressure function. As a consequence, we show that the Hausdorff, packing and box counting dimensions of such systems are equal to the zero of the pressure. In particular, we do not presume the validity of the Falconer-Sloan condition or any other additional assumptions which have been essential in all the previously known results.
Let $μ$ be a self-affine measure on $\mathbb{R}^{d}$ associated to a self-affine IFS $\{φ_λ(x) = A_λx + v_λ\}_{λ\inΛ}$ and a probability vector $p=(p_λ)_λ>0$. Assume the strong separation condition holds. Let $γ_{1}\ge...\geγ_{d}$ and $D$ be the Lyapunov exponents and dimension corresponding to $\{A_λ\}_{λ\inΛ}$ and $p^{\mathbb{N}}$, and let $\mathbf{G}$ be the group generated by $\{A_λ\}_{λ\inΛ}$. We show that if $γ_{m+1}>γ_{m}=...=γ_{d}$, if $\mathbf{G}$ acts irreducibly on the vector space of alternating $m$-forms, and if the Furstenberg measure $μ_{F}$ satisfies $\dim_{H}μ_{F}+D>(m+1)(d-m)…
Hausdorff Dimension of a Class of Self-Affine Sets
APJ Abdul Kalam Technological University（阿卜杜勒·卡拉姆理工大学）；Rajagiri School of Engineering and Technology（拉吉里工程技术学院）；Muthoot Institute of Technology and Science（穆托特科技与理工学院）
本文推导了由仿射迭代函数系统生成的自仿集类的精确Hausdorff维数公式，结合了收缩相似变换与对称算子A^T A的线性部分，统一了混合系统的方法，证明了参数平衡c + r = 1对吸引子连通性和开放集条件的保证。
An iterated function system $Φ$ consisting of contractive similarity mappings has a unique attractor $F \subseteq \mathbb{R}^d$ which is invariant under the action of the system, as was shown by Hutchinson [Hut]. This paper shows how the action of the function system naturally produces a tiling $\mathcal{T}$ of the convex hull of the attractor. More precisely, it tiles the complement of the attractor within its convex hull. These tiles form a collection of sets whose geometry is typically much simpler than that of $F$, yet retains key information about both $F$ and $Φ$. In particular, the tile…
Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine subsets may be chosen so as to have stronger separation properties and in such a way that the linear parts of their affinities are positive matrices. Combining this result with some recent breakthroughs in the study of self-affine measures and their associated Furstenberg measures, we obtain new criteria under which the Hausdorff dimension of a self-affine …
Let $I=[0,1)$, $-1<λ<1$ and $f\colon I\to I$ be a piecewise $λ$-affine map of the interval $I$, i.e., there exist a partition $0=a_0<a_1<\cdots< a_{k-1}<a_k=1$ of the interval $I$ into $k\geq2$ subintervals and $b_1,\ldots, b_k\in\mathbb{R}$ such that $f(x)=λx+ b_i$ for every $x\in[a_{i-1},a_{i})$ and $i=1,\ldots,k$. The exceptional set $\mathcal{E}_f$ of $f$ is the set of parameters $δ\in\mathbb{R}$ such that $R_δ\circ f$ is not asymptotically periodic, where $R_δ\colon I\to I$ is the rotation of angle $δ$. In this paper we prove that $\mathcal{E}_f$ has zero Hausdorff dimension. We derive th…
We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. In particular, for self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both …
Overlap coincidence in a self-affine tiling in $\R^d$ is equivalent to pure point dynamical spectrum of the tiling dynamical system. We interpret the overlap coincidence in the setting of substitution Delone set in $\R^d$ and find an efficient algorithm to check the pure point dynamical spectrum. This algorithm is easy to implement into a computer program. We give the program and apply it to several examples. In the course the proof of the algorithm, we show a variant of the conjecture of Urbański (Solomyak \cite{Solomyak:08}) on the Hausdorff dimension of the boundaries of fractal tiles.
In 1988 K. Falconer introduced a formula which predicts the value of the Hausdorff dimension of the attractor of an affine iterated function system. The value given by this formula -- sometimes referred to as the affinity dimension -- is known to agree with the Hausdorff dimension both generically and in an increasing range of explicit cases. It is however a nontrivial problem to estimate the numerical value of the affinity dimension for specific iterated function systems. In this article we substantially extend an earlier result of M. Pollicott and P. Vytnova on the computation of the affinit…
The connected part of a metric space $E$ is defined to be the union of non-trivial connected components of $E$. We proved that for a class of self-affine sets called slicing self-affine sponges, the connected part of $E$ either coincides with $E$, or is essentially contained in the attractor of a proper sub-IFS of an iteration of the original IFS.This generalize an early result of Huang and Rao [L. Y. Huang, H. Rao. \emph{A dimension drop phenomenon of fractal cubes}, J. Math. Anal. Appl. \textbf{497} (2021), no. 2] on a class of self-similar sets called fractal cubes. Moreover, we show that t…
For a decreasing real valued function $ψ$, a pair $(A,\mathbf{b})$ of a real $m\times n$ matrix $A$ and $\mathbf{b}\in\mathbb{R}^m$ is said to be $ψ$-Dirichlet improvable if the system $$\|A\mathbf{q}+\mathbf{b}-\mathbf{p}\|^m < ψ(T)\quad\text{and}\quad\|\mathbf{q}\|^n < T$$ has a solution $\mathbf{p}\in\mathbb{Z}^m$, $\mathbf{q}\in\mathbb{Z}^n$ for all sufficiently large $T$, where $\|\cdot\|$ denotes the supremum norm. Kleinbock and Wadleigh (2019) established an integrability criterion for the Lebesgue measure of the $ψ$-Dirichlet non-improvable set. In this paper, we prove a similar criter…
Let $f$ and $g$ be transcendental entire functions, each with a bounded set of singular values, and suppose that $f$ and $g$ are affinely equivalent (that is, $g \circ ϕ= ψ\circ f$, where $ϕ,ψ:\C\to\C$ are affine). We show that the escaping sets of $f$ and $g$ have the same Hausdorff dimension. Using a result of the second author, we deduce that there exists a family of transcendental entire functions for which the escaping set has Hausdorff dimension equal to one.
A well-known theorem of J.E. Hutchinson states that if an iterated function system consists of similarity transformations and satisfies the open set condition then its attractor supports a self-similar measure with Hausdorff dimension equal to the similarity dimension. In this article we prove the following result which may be regarded as a form of partial converse: if an iterated function system consists of invertible affine transformations whose linear parts do not preserve a common invariant subspace, and its attractor supports a self-affine measure with Hausdorff dimension equal to the aff…
Budapest University of Technology and Economics（布达佩斯技术与经济大学）；HUN-REN-BME Stochastics Research Group（匈牙利研究与教育网络-布达佩斯技术与经济大学随机过程研究组）；Institute of Mathematics（数学研究所）
Let E be a plane self-affine set defined by affine transformations with linear parts given by matrices with positive entries. We show that if mu is a Bernoulli measure on E with dim_H mu = dim_L mu, where dim_H and dim_L denote Hausdorff and Lyapunov dimensions, then the projection of mu in all but at most one direction has Hausdorff dimension min{dim_H mu,1}. We transfer this result to sets and show that many self-affine sets have projections of dimension min{dim_H E,1} in all but at most one direction.
We study the surface regularity of compact sets $G \subset R^n$ which is equal to the supremum of numbers $s\ge 0$ such that the measure of the set $G_{\varepsilon}\setminus G$ does not exceed $C\varepsilon^{s}, \varepsilon > 0$, where $G_{\varepsilon}$ denotes the $\varepsilon$-neighbourhood of~$G$. The surface dimension is by definition the difference between~$n$ and the surface regularity. Those values provide a natural characterisation of regularity for sets of positive measure. We show that for self-affine attractors and tiles those characteristics are explicitly computable and find them …
Let $G$ be a group and $ϕ:H\to G$ be a contracting homomorphism from a subgroup $H<G$ of finite index. V.Nekrashevych [25] associated with the pair $(G,ϕ)$ the limit dynamical system $(\lims,\si)$ and the limit $G$-space $\limGs$ together with the covering $\cup_{g\in G}\tile\cdot g$ by the tile $\tile$. We develop the theory of self-similar measures $μ$ on these limit spaces. It is shown that $(\lims,\si,μ)$ is conjugated to the one-sided Bernoulli shift. Using sofic subshifts we prove that the tile $\tile$ has integer measure and we give an algorithmic way to compute it. In addition we give …
In this paper, we study cut sets of attractors of iteration function systems (IFS) in $\mathbb{R}^d$. Under natural conditions, we show that all irreducible cut sets of these attractors are perfect sets or single points. This leads to a criterion for the existence of cut points of IFS attractors. If the IFS attractors are self-affine tiles, our results become algorithmically checkable and can be used to exhibit cut points with the help of Hata graphs. This enables us to construct cut points of some self-affine tiles studied in the literature.
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. Käenmäki 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 exam…
In this paper we compute the dimension of a class of dynamically defined non-conformal sets. Let $X\subseteq\mathbb{T}^2$ denote a Bedford-McMullen set and $T:X\to X$ the natural expanding toral endomorphism which leaves $X$ invariant. For an open set $U\subset X$ we let X_U={x\in X : T^k(x)\not\in U \text{for all}k}. We investigate the box and Hausdorff dimensions of $X_U$ for both a fixed Markov hole and also when $U$ is a shrinking metric ball. We show that the box dimension is controlled by the escape rate of the measure of maximal entropy through $U$, while the Hausdorff dimension depends…
In this paper we consider diagonally affine, planar IFS $Φ=\left\{S_i(x,y)=(α_ix+t_{i,1},β_iy+t_{i,2})\right\}_{i=1}^m$. Combining the techniques of Hochman and Feng, Hu we compute the Hausdorff dimension of the self-affine attractor and measures and we give an upper bound for the dimension of the exceptional set of parameters.
We show that in a typical sub-self-affine set, the Hausdorff and the Minkowski dimensions coincide and equal the zero of an appropriate topological pressure. This gives a partial positive answer to the question of Falconer. We also study the properties of the topological pressure and the existence and the uniqueness of natural measures supported on a sub-self-affine set.
We develop a thermodynamic formalism for quasi-multiplicative potentials on a countable symbolic space and apply these results to the dimension theory of infinitely generated self-affine sets. The first application is a generalisation of Falconer's dimension formula to include typical infinitely generated self-affine sets and show the existence of an ergodic invariant measure of full dimension whenever the pressure function has a root. Considering the multifractal analysis of Birkhoff averages of general potentials $Φ$ taking values in $\R^{\N}$, we give a formula for the Hausdorff dimension o…
We prove that for all $b$, the Hausdorff dimension of the set of $m \times n$ matrices $ε$-badly approximable for the target $b$ is not full. The doubly metric case follows. It was known that for almost every matrix $A$, the Hausdorff dimension of the set $Bad_A(ε)$ of $ε$-badly approximable target $b$ is not full, and that for real numbers $α$, $\dim_H Bad_α(ε)=1$ if and only if $α$ is singular on average. We show that if $\dim_H Bad_A(ε)=m$, then $A$ is singular on average.
We prove that generically, for a self-affine set in $\mathbb{R}^d$, removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension. This gives a partial positive answer to a folklore open question.
In this note we consider the Hausdorff dimension of self-affine sets with random perturbations. We extend previous work in this area by allowing the random perturbation to be distributed according to distributions with unbounded support as long as the measure of the tails of the distribution decay super polynomially.
