Starting from the random extension of the Cantor middle set in [0,1], by iteratively removing the central uniform spacing from the intervals remaining in the previous step, we define random Beta(p,1)-Cantor sets, and compute their Hausdorff dimension. Next we define a deterministic counterpart, by iteratively removing the expected value of the spacing defined by the appropriate Beta(p,1) order statistics. We investigate the reasons why the Hausdorff dimension of this deterministic fractal is greater...

Consider a graph directed iterated function system (GIFS) on the line which consists of similarities. Assuming neither any separation conditions, nor any restrictions on the contractions, we compute the almost sure dimension of the attractor. Then we apply our result to give a partial answer to an open problem in the field of fractal image recognition concerning some self-affine graph directed attractors in space.

For any $x\in (0,1]$, let $$x=\frac{1}{d_1}+\frac{1}{d_1(d_1-1)d_2}+\cdots +\frac{1}{d_1(d_1-1)\cdots d_{n-1}(d_{n-1}-1)d_n}+\cdots$$ be its Lüroth expansion. Denote by $P_n\left(x\right)/Q_n\left(x\right)$ the partial sum of the first $n$ terms in the above series and call it the $n$th convergent of $x$ in the Lüroth expansion. This paper is concerned with the efficiency of approximating real numbers by their convergents ${\{P_n\left(x\right)/Q_n\left(x\right)\}}_{n\ge 1}$ in the Lüroth expansion. It is shown that almost no points can have convergents as the optimal approximation for infinitely many times in the Lüroth expansion. Consequently, Hausdorff dimension is introduced to quantify the set of...

It is well known that every $x\in (0,1]$ can be expanded to an infinite Lüroth series in the form of $$x=\frac{1}{d_1\left(x\right)}+\cdots +\frac{1}{d_1\left(x\right)(d_1\left(x\right)-1)\cdots d_{n-1}\left(x\right)(d_{n-1}\left(x\right)-1)d_n\left(x\right)}+\cdots ,$$ where $d_n\left(x\right)\ge 2$ for all $n\ge 1$. In this paper, sets of points with some restrictions on the digits in Lüroth series expansions are considered. Mainly, the Hausdorff dimensions of the Cantor sets $$F_{\phi }=\{x\in (0,1]:d_n\left(x\right)\ge \phi \left(n\right),\forall n\ge 1\}$$ are completely determined, where $\phi$ is an integer-valued function defined on $\mathbb{N}$, and $\phi \left(n\right)\to \infty$ as $n\to \infty$.

It is known that with a non-unit Pisot substitution $\sigma$ one can associate certain fractal tiles, so-called Rauzy fractals. In our setting, these fractals are subsets of a certain open subring of the adèle ring of the associated Pisot number field. We present several approaches on how to define Rauzy fractals and discuss the relations between them. In particular, we consider Rauzy fractals as the natural geometric objects of certain numeration systems, in terms of the dual of the one-dimensional realization...

Let ${fₙ}_{n\ge 1}$ be an infinite iterated function system on [0,1] satisfying the open set condition with the open set (0,1) and let Λ be its attractor. Then to any x ∈ Λ (except at most countably many points) corresponds a unique sequence ${aₙ\left(x\right)}_{n\ge 1}$ of integers, called the digit sequence of x, such that $x=li{m}_{n\to \infty }{f}_{a₁\left(x\right)}\circ \cdots \circ f_{aₙ\left(x\right)}\left(1\right)$. We investigate the growth speed of the digits in a general infinite iterated function system. More precisely, we determine the dimension of the set $x\in \Lambda :aₙ\left(x\right)\in B\left(\forall n\ge 1\right),li{m}_{n\to \infty }aₙ\left(x\right)=\infty$ for any infinite subset B ⊂ ℕ, a question posed by Hirst for continued...

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.

Very recently bounds for the L q spectra of inhomogeneous self-similar measures satisfying the Inhomogeneous Open Set Condition (IOSC), being the appropriate version of the standard Open Set Condition (OSC), were obtained. However, if the IOSC is not satisfied, then almost nothing is known for such measures. In the paper we study the L q spectra and Rényi dimension of generalized inhomogeneous self-similar measures, for which we allow an infinite number of contracting similarities and probabilities...

We compute the typical (in the sense of Baire’s category theorem) multifractal box dimensions of measures on a compact subset of ${ℝ}^d$. Our results are new even in the context of box dimensions of measures.