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collage theorem

{{short description|Characterises an iterated function system whose attractor is close to a given set}}

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.

See also

References

  • {{cite book

| author = Barnsley, Michael.

| title = Fractals Everywhere

| url = https://archive.org/details/fractalseverywhe0000barn

| url-access = registration

| year = 1988

| publisher = Academic Press, Inc.

| isbn = 0-12-079062-9

}}

External links

  • [http://www.cut-the-knot.org/ctk/ifs.shtml A description of the collage theorem and interactive Java applet] at cut-the-knot.
  • [https://web.archive.org/web/20080917024731/http://www.math.okstate.edu/mathdept/dynamics/lecnotes/node47.html Notes on designing IFSs to approximate real images.]
  • [https://web.archive.org/web/20100606074948/http://scimath.unl.edu/MIM/files/MATExamFiles/Snyder_MAT_Exam_ExpositoryPaper.pdf Expository Paper on Fractals and Collage theorem]

{{DEFAULTSORT:Collage Theorem}}

Category:Fractals

Category:Theorems in geometry

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