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.

Statement
Let $$\mathbb{X}$$ be a complete metric space. Suppose $$L$$ is a nonempty, compact subset of $$\mathbb{X}$$ and let $$\epsilon >0$$ be given. Choose an iterated function system (IFS) $$\{ \mathbb{X} ; w_1, w_2, \dots, w_N\}$$ with contractivity factor $$s,$$ where $$0 \leq s < 1$$ (the contractivity factor $$s$$ of the IFS is the maximum of the contractivity factors of the maps $$w_i$$). Suppose


 * $$h\left( L, \bigcup_{n=1}^N w_n (L) \right) \leq \varepsilon,$$

where $$h(\cdot,\cdot)$$ is the Hausdorff metric. Then


 * $$h(L,A) \leq \frac{\varepsilon}{1-s}$$

where A is the attractor of the IFS. Equivalently,


 * $$h(L,A) \leq (1-s)^{-1} h\left(L,\cup_{n=1}^N w_n(L)\right) \quad$$, for all nonempty, compact subsets L of $$\mathbb{X}$$.

Informally, If $$L$$ is close to being stabilized by the IFS, then $$L$$ is also close to being the attractor of the IFS.