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Fractals in the large. (English) Zbl 0913.28005
The paper studies the fractal structures at large scales in two ways: reverse iterated function systems (RIFS) and fractal blowups. A reverse iterated function system is defined to be a set of expansive maps \(\{T_1,T_2,\dots, T_m\}\) on a discrete metric space. A set \(F\) is called an invariant set of \(\{T_1,T_2,\dots, T_m\}\) if \(F= \sum^m_{j=1} T_jF\) and a measure \(\mu\) an invariant measure if \(\mu\) is the solution of \(\mu= \sum^m_{j= 1} p_j\mu\circ T^{-1}_j\) for positive weights \(p_j\). The structures and basic properties of such invariant sets and measures are investigated. It is proved that invariant sets can be described as unions of forward orbits of fixed points of iterated maps from the RIFS. A blowup \({\mathcal F}\) of a self-similar set \(F\) in \(\mathbb{R}^n\) is defined to be the union of an increasing sequence of sets, each similar to \(F\). A general construction of blowups is given and the blowups of the Cantor set, the Sierpiński gasket, and the von Koch curve are described.

MSC:
28A80 Fractals
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