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.
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.
