proof of existence and unicity of self-similar fractals

We consider the space 𝒦(X)={KX:Kcompactandnonempty} endowed with the Hausdorff distance δ. Since Hausdorff metric inherits completeness, being X completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, (𝒦(X),δ) is complete too. We then consider the mapping T:𝒦(X)𝒦(X) defined by


We claim that T is a contraction. In fact, recalling that δ(A1A2,B1B2)max{δ(A1,B1),δ(A2,B2)} while δ(Ti(A),Ti(B))λiδ(A,B) if Ti is λi-LipschitzPlanetmathPlanetmath, we have

δ(T(A),T(B)) =δ(iTi(A),iTi(B))maxiδ(Ti(A),Ti(B))

with λ=maxiλi<1.

So T is a contraction on the complete metric space 𝒦(X) and hence, by Banach Fixed Point TheoremMathworldPlanetmath, there exists one and only one K𝒦(X) such that T(K)=K.

Title proof of existence and unicity of self-similar fractals
Canonical name ProofOfExistenceAndUnicityOfSelfsimilarFractals
Date of creation 2013-03-22 16:05:30
Last modified on 2013-03-22 16:05:30
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 5
Author paolini (1187)
Entry type Proof
Classification msc 28A80