Related papers: Boundaries of Disk-like Self-affine Tiles

In this paper, we study the local connectivity and Hausdorff dimension for the boundaries of the bounded hyperbolic components in the space $\mathcal P_d$ of polynomials of degree $d\geq 3$. It is shown that for any non disjoint-type…

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…

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…

In this paper, exact Hausdorff dimension formulas for a class of self-affine attractors generated by affine Iterated Function Systems are derived. We consider systems containing an affine map whose $n$-th iterate is a similarity…

Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\|<1/2$ for $1\leq i\leq m$. For ${\bf a}=(a_1,\ldots, a_m)\in \Bbb R^{md}$, let $\pi^{{\bf a}}:\; \Sigma=\{1,\ldots, m\}^{\Bbb N}\to \Bbb R^d$ denote the…

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…

We prove that a compact metric space (or more generally an analytic subset of a complete separable metric space) of Hausdorff dimension bigger than $k$ can be always mapped onto a $k$-dimensional cube by a Lipschitz map. We also show that…

Given a compact K\"ahler manifold $(X,\omega)$, due to the work of Darvas-Di Nezza-Lu, the space of singularity types of $\omega$-psh functions admits a natural pseudo-metric $d_\mathcal S$ that is complete in the presence of positive mass.…

Let $D\geq 3$ denote an integer. For any $x\in \mathbb F_2^D$ let $w(x)$ denote the Hamming weight of $x$. Let $X$ denote the subspace of $\mathbb F_2^D$ consisting of all $x\in \mathbb F_2^D$ with even $w(x)$. The $D$-dimensional halved…

Supersymmetric domain-wall spacetimes that lift to Ricci-flat solutions of M-theory admit generalized Heisenberg (2-step nilpotent) isometry groups. These metrics may be obtained from known cohomogeneity one metrics of special holonomy by…

For solutions to the $\mathfrak{g}_2$ affine Toda field equations in $\mathbb{C}$ with respect to \emph{polynomial} holomorphic sextic differential $q$, we study the associated almost-complex curves $\nu_q: \mathbb{C} \rightarrow…

Note by the author: Section 9.3 is added from the more general unpublished manuscript ``A Perturbation Method Leading to Full-Dimension Ergodic Measures on Integral Self-Affine Sets'', (2021) by I. Kirat. Original abstract: An integral…

In this work, we present a new method for generating a threshold structure. This kind of structure can be advantageously used in various halftoning algorithms such as clustered-dot or dispersed-dot dithering, error diffusion with threshold…

In this paper, we consider homogeneous $\Delta_H$-harmonic polynomials on the first Heisenberg group $\mathbb H$ and their traces on the unit sphere $S_\rho$ associated with the Kor\'anyi--Folland homogeneous norm $\rho$. We prove that…

A theorem of Harald Bohr (1914) states that if f is a holomorphic map from the unit disc into itself, then the sum of absolute values of its Taylor expansion is less than 1 for |z|<1/3. The bound 1/3 is optimal. This result has been…

For countably infinite IFSs on $\mathbb R^2$ consisting of affine contractions with diagonal linear parts, we give conditions under which the affinity dimension is an upper bound for the Hausdorff dimension and a lower bound for the lower…

Let $A$ be an Artinian Gorenstein algebra over an infinite field $k$ with either $\hbox{char}(k)=0$ or $\hbox{char}(k)>\nu$, where $\nu$ is the socle degree of $A$. To every such algebra and a linear projection $\pi$ on its maximal ideal…

Fino and Kath determined all possible holonomy groups of seven-dimensional pseu\-do-Rie\-man\-nian manifolds contained in the exceptional, non-compact, simple Lie group $\mathrm{G}_2^*$ via the corresponding Lie algebras. They are…

Let $f$ be a polynomial in the free algebra over a field $K$, and let $A$ be a $K$-algebra. We denote by $\S_A(f)$, $\A_A(f)$ and $\I_A(f)$, respectively, the `verbal' subspace, subalgebra, and ideal, in $A$, generated by the set of all…

Let $A$ be an expanding $d\times d$ matrix with integer entries and ${\mathcal D}\subset {\mathbb Z}^d$ be a finite digit set. Then the pair $(A, {\mathcal D})$ defines a unique integral self-affine set $K=A^{-1}(K+{\mathcal D})$. In this…