Английская Википедия:Collage theorem

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

<math>h\left( L, \bigcup_{n=1}^N w_n (L) \right) \leq \varepsilon,</math>

where <math>h(\cdot,\cdot)</math> is the Hausdorff metric. Then

<math>h(L,A) \leq \frac{\varepsilon}{1-s}</math>

where A is the attractor of the IFS. Equivalently,

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

Informally, If <math>L</math> is close to being stabilized by the IFS, then <math>L</math> is also close to being the attractor of the IFS.

See also

• Michael Barnsley
• Barnsley fern

References

• Шаблон:Cite book

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

Шаблон:Fractal-stub