For $\lambda \in (0,\,1/2]$ let $K_\lambda \subset \mathbb {R}$ be a self-similar set generated by the iterated function system $\{\lambda x,\, \lambda x+1-\lambda \}$. Given $x\in (0,\,1/2)$, let $\Lambda (x)$ be the set of $\lambda \in (0,\,1/2]$ such that $x\in K_\lambda$. In this paper we show that $\Lambda (x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any $y_1,\,\ldots,\, y_p\in (0,\,1/2)$ there exists a full Hausdorff dimensional set of $\lambda \in (0,\,1/2]$ such that $y_1,\,\ldots,\, y_p \in K_\lambda$.

Let $K\subset {\mathbb {R}}^d$ be a self-similar set generated by an iterated function system $\{\varphi _i\}_{i=1}^m$ satisfying the strong separation condition and let f be a contracting similitude with $f(K)\subseteq K$. We show that $f(K)$ is relatively open in K if all $\varphi _i$ share a common contraction ratio and orthogonal part. We also provide a counterexample when the orthogonal parts are allowed to vary. This partially answers a question of Elekes, Keleti and Máthé [Ergod. Th. & Dynam. Sys. 30 (2010), 399–440]. As a byproduct of our argument, when $d=1$ and K admits two homogeneous generating iterated function systems satisfying the strong separation condition but with contraction ratios of opposite signs, we show that K is symmetric. This partially answers a question of Feng and Wang [Adv. Math. 222 (2009), 1964–1981].

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

Croft, Falconer and Guy asked: what is the smallest integer $n$ such that an $n$-reptile in the plane has a hole? Motivated by this question, we describe a geometric method of constructing reptiles in $\mathbb{R}^d$, especially reptiles with holes. In particular, we construct, for each even integer $n\ge4$, an $n$-reptile in $\mathbb{R}^2$ with holes. We also answer some questions concerning the topological properties of a reptile whose interior consists of infinitely many components.

An n-hedral tiling of ℝd is a tiling with each tile congruent to one of the n distinct sets. In this paper, we use the iterated function systems (IFS) to generate n-hedral tilings of ℝd. Each tile in the tiling is similar to the attractor of the IFS. These tiles are fractals and their boundaries have the Hausdorff dimension less than d. Our results generalize a result of Bandt.