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 $\varepsilon >0$ 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,

$h(L,A)\leq (1-s)^{-1}h\left(L,\bigcup _{n=1}^{N}w_{n}(L)\right)$

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.

See also

• Michael Barnsley
• Barnsley fern

References

• Barnsley, Michael. (1988). Fractals Everywhere. Academic Press, Inc. ISBN 0-12-079062-9.

External links

• A description of the collage theorem and interactive Java applet at cut-the-knot.
• Notes on designing IFSs to approximate real images.
• Expository Paper on Fractals and Collage theorem