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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].

Summary

One of the most important and widely used methods for constructing fractal sets is via iterated function systems. These were introduced in a celebrated article of Hutchinson from 1980. We consider the special case of self-similar sets in Chapter 7, self-affine sets in Chapter 8, and self-conformal sets in Section 9.1, but in this chapter we introduce the theory of iterated function systems and their attractors in full generality.

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On a self-embedding problem for self-similar sets

6 - Iterated Function Systems

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On a self-embedding problem for self-similar sets

6 - Iterated Function Systems

Summary