Dixmier trace

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 of the theorem

Let $\mathbb{𝕏}$ be a complete metric space. Let $L\in H(\mathbb{𝕏})$ be given, and let $ϵ\ge 0$ be given. Choose an iterated function system (IFS) $\{\mathbb{𝕏};w_1,w_2,\dots ,w_N\}$ with contractivity factor $0\le s<1$, so that

$h(L,\bigcup _{n=1}^N{w}_n(L))\le \epsilon ,$

where $h(d)$ is the Hausdorff metric. Then

$h(L,A)\le \frac{\epsilon }{1-s}$

where A is the attractor of the IFS. Equivalently,

$h(L,A)\le (1-s{)}^{-1}h(L,\underset{n=1}{\overset{N}{\cup }}{w}_n(L))\text{for all }L\in H(\mathbb{𝕏}).$

See also

• Michael Barnsley

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

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

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