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 $\epsilon \geq 0$ be given. Choose an iterated function system (IFS) $\{\mathbb {X} ;w_{1},w_{2},\dots ,w_{N}\}$ with contractivity factor $0\leq s<1$, so that

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

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

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

where A is the attractor of the IFS. Equivalently,

$h(L,A)\leq (1-s)^{-1}h\left(L,\cup _{n=1}^{N}w_{n}(L)\right)\quad {\text{for all }}L\in {H}(\mathbb {X} ).$

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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