Attractors for iterated function systems

Abstract

Let $X$ be a compact metric space with $\mathcal{S}=\{S_1,\dots ,S_N\}$ a finite set of contraction maps from $X$ to itself. Call a subset $F$ of $X$ an attractor for the iterated function scheme (IFS) $\mathcal{S}$ if $F={\bigcup }_{i=1}^N{S}_i(F)$. Working primarily on the unit interval $I=[0,1]$, we show that

1. the typical closed set in $[0,1]$ is not an attractor of an IFS, and describe the closed sets that comprise the $F_{\sigma }$ set of attractors;
2. both the set of attractors and its complement are dense subsets of $(\mathcal{K}([0,1]),\mathcal{H})$;
3. the set of attractors is path-connected;
4. every countable compact subset of $[0,1]$ of finite Cantor–Bendixon rank is homeomorphic to an attractor, and
5. every nowhere dense uncountable compact subset of $[0,1]$ is homeomorphic to an attractor.