proof of existence and unicity of self-similar fractals

We consider the space $𝒦(X)=\{K\subset X:K\mathrm{compact}\mathrm{and}\mathrm{non}\mathrm{empty}\}$ endowed with the Hausdorff distance $\delta$. Since Hausdorff metric inherits completeness, being $X$ complete, $(𝒦(X),\delta )$ is complete too. We then consider the mapping $T:𝒦(X)\to 𝒦(X)$ defined by

$$T(A)=\bigcup _{i=1}^N{T}_i(A).$$

We claim that $T$ is a contraction. In fact, recalling that $\delta (A_1\cup A_2,B_1\cup B_2)\le \max \{\delta (A_1,B_1),\delta (A_2,B_2)\}$ while $\delta (T_i(A),T_i(B))\le {\lambda }_i\delta (A,B)$ if $T_i$ is ${\lambda }_i$-Lipschitz, we have

$\delta (T(A),T(B))$ $=\delta ({\displaystyle \bigcup _i}{T}_i(A),{\displaystyle \bigcup _i}{T}_i(B))\le \underset{i}{\max }\delta (T_i(A),T_i(B))$

$\le \underset{i}{\max }{\lambda }_i\delta (A,B)=\lambda \delta (A,B)$

with .

So $T$ is a contraction on the complete metric space $𝒦(X)$ and hence, by Banach Fixed Point Theorem, there exists one and only one $K\in 𝒦(X)$ 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