Collage theorem

In mathematics, the collage theorem characterises an iterated function system whose attractor is close, relative to the Hausdorff metric, to a given set. The IFS described is composed of contractions whose images, as a collage or union when mapping the given set, are arbitrarily close to the given set. It is typically used in fractal compression.

Let ${\displaystyle \mathbb {X} }$ be a complete metric space. Suppose ${\displaystyle L}$ is a nonempty, compact subset of ${\displaystyle \mathbb {X} }$ and let ${\displaystyle \epsilon \geq 0}$ be given. Choose an iterated function system (IFS) ${\displaystyle \{\mathbb {X} ;w_{1},w_{2},\dots ,w_{N}\}}$ with contractivity factor ${\displaystyle 0\leq s<1}$, (The contractivity factor of the IFS is the maximum of the contractivity factors of the maps ${\displaystyle w_{i}}$.) Suppose

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