Related papers: Multiple codings for self-similar sets with overla…

Let $\lambda_{1},\ldots,\lambda_{n}$ be real numbers in $(0,1)$ and $p_{1},\ldots,p_{n}$ be points in $\mathbb{R}^{d}$. Consider the collection of maps $f_{j}:\mathbb{R}^{d}\to\mathbb{R}^{d} $ given by $$f_{j}(x)=\lambda_{j} x…

This paper studies a large class of continuous functions $f:[0,1]\to\mathbb{R}^d$ whose range is the attractor of an iterated function system $\{S_1,\dots,S_{m}\}$ consisting of similitudes. This class includes such classical examples as…

We formulate the weak separation condition and the finite type condition for conformal iterated function systems on Riemannian manifolds with nonnegative Ricci curvature, and generalize the main theorems by Lau \textit{et al.} in [Monatsch.…

Let $\Phi$ be a $C^{1+\gamma}$ smooth IFS on $\mathbb{R}$, where $\gamma>0$. We provide mild conditions on the derivative cocycle that ensure that every self conformal measure is supported on points $x$ that are absolutely normal. That is,…

In this article we calculate the Hausdorff dimension of the set \begin{equation*} \mathcal{F}(\Phi )=\left\{ x\in \lbrack 0,1):\begin{aligned}a_{n+1}(x)a_n(x) \geq \Phi(n) \ {\rm for \ infinitely \ many \ } n\in \mathbb N \ {\rm and } \\…

We study invariant measures for random countable (finite or infinite) conformal iterated function systems (IFS) with arbitrary overlaps. We do not assume any type of separation condition. We prove, under a mild assumption of finite entropy,…

Given any dimension function $h$, we construct a perfect set $E \subseteq \mathbb{R}$ of zero $h$-Hausdorff measure, that contains any finite polynomial pattern. This is achieved as a special case of a more general construction in which we…

For any family of measurable sets in a probability space, we show that either (i) the family has infinite Vapnik-Chervonenkis (VC) dimension or (ii) for every epsilon > 0 there is a finite partition pi such the pi-boundary of each set has…

This work is devoted to the study of families of infinite parabolic iterated function systems (PIFS) on a closed interval parametrized by vectors in $\mathbb{R}^d$ with overlaps. We show that the Hausdorff dimension and absolute continuity…

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…

It is known that, unlike the Hausdorff dimension, the Assouad dimension of a self-similar set can exceed the similarity dimension if there are overlaps in the construction. Our main result is the following precise dichotomy for self-similar…

We introduce the concept of F-decomposable systems, well-ordered inverse systems of Hausdorff compacta with fully closed bonding mappings. A continuous mapping between Hausdorff compacta is called fully closed if the intersection of the…

We show that any equicontractive, self-similar measure arising from the IFS of contractions $(S_{j})$, with self-similar set $[0,1]$, admits an isolated point in its set of local dimensions provided the images of $S_{j}(0,1)$ (suitably)…

Until recently, it was an important open problem in Fractal Geometry to determine whether there exists an iterated function system acting on $\mathbb{R}$ with no exact overlaps for which cylinders are super-exponentially close at all small…

In this note we present some one-parameter families of homogeneous self-similar measures on the line such that - the similarity dimension is greater than $1$ for all parameters and - the singularity of some of the self-similar measures from…

Suppose $n\geq 2$ and $\mathcal{A}_{i}\subset \{0,1,\cdots ,(n-1)\}$ for $ i=1,\cdots ,l,$ let $K_{i}=\bigcup\nolimits_{a\in \mathcal{A}_{i}}n^{-1}(K_{i}+a)$ be self-similar sets contained in $[0,1].$ Given $ m_{1},\cdots ,m_{l}\in…

In the setting of a metric space equipped with a doubling measure that supports a Poincar\'e inequality, we show that a set $E$ is of finite perimeter if and only if $\mathcal H(\partial^1 I_E)<\infty$, that is, if and only if the…

Let $\bm p_0,...,\bm p_{m-1}$ be points in ${\mathbb R}^d$, and let $\{f_j\}_{j=0}^{m-1}$ be a one-parameter family of similitudes of ${\mathbb R}^d$: $$ f_j(\bm x) = \lambda\bm x + (1-\lambda)\bm p_j, j=0,...,m-1, $$ where…

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…

We study the singularity (multifractal) spectrum of continuous convex functions defined on $[0,1]^{d}$. Let $E_f({h}) $ be the set of points at which $f$ has a pointwise exponent equal to $h$. We first obtain general upper bounds for the…