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# Fractals in the large

Published:1998-06-01
Printed: Jun 1998
• Robert S. Strichartz
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## Abstract

A {\it reverse iterated function system} (r.i.f.s.) is defined to be a set of expansive maps $\{T_1,\ldots,T_m\}$ on a discrete metric space $M$. An invariant set $F$ is defined to be a set satisfying $F = \bigcup^m_{j=1} T_jF$, and an invariant measure $\mu$ is defined to be a solution of $\mu = \sum^m_{j=1} p_j\mu\circ T_j^{-1}$ for positive weights $p_j$. The structure and basic properties of such invariant sets and measures is described, and some examples are given. A {\it blowup} $\cal F$ of a self-similar set $F$ in $\Bbb R^n$ is defined to be the union of an increasing sequence of sets, each similar to $F$. We give a general construction of blowups, and show that under certain hypotheses a blowup is the sum set of $F$ with an invariant set for a r.i.f.s. Some examples of blowups of familiar fractals are described. If $\mu$ is an invariant measure on $\Bbb Z^+$ for a linear r.i.f.s., we describe the behavior of its {\it analytic} transform, the power series $\sum^\infty_{n=0} \mu(n)z^n$ on the unit disc.