Puneet Varma (Editor)

Collage theorem

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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 X be a complete metric space. Suppose L is a nonempty, compact subset of X and let ϵ 0 be given. Choose an iterated function system (IFS) { X ; w 1 , w 2 , , w N } with contractivity factor 0 s < 1 , (The contractivity factor of the IFS is the maximum of the contractivity factors of the maps w i .) Suppose

h ( L , n = 1 N w n ( L ) ) ε ,

where h ( , ) is the Hausdorff metric. Then

h ( L , A ) ε 1 s

where A is the attractor of the IFS. Equivalently,

h ( L , A ) ( 1 s ) 1 h ( L , n = 1 N w n ( L ) ) , for all nonempty, compact subsets L of X .

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

References

Collage theorem Wikipedia