# self-similar fractals

Let $(X,d)$ be a metric space and let $T_{1},\ldots,T_{N}$ be a finite number of contractions on $X$ i.e. each $T_{i}\colon X\to X$ enjoys the property

 $d(T_{i}(x),T_{i}(y))\leq\lambda_{i}d(x,y)$

($T_{i}$ is $\lambda_{i}$-Lipschitz) for some $\lambda_{i}<1$.

Given a set $A\subset X$ we can define

 $T(A)=\bigcup_{i=1}^{N}T_{i}(A).$
###### Definition 1.

A set $K$ such that $T(K)=K$ (invariant set) is called a self-similar fractal with respect to the contractions $\{T_{1},\ldots,T_{N}\}$.

A more interesting example is the Koch curve  in $X=\mathbb{R}^{2}$. In this case we choose $N=4$ similitudes with factor $1/3$.

###### Theorem 1.

Let $X$ be a complete metric space and let $T_{1},\ldots,T_{N}\colon X\to X$ be a given set of contractions. Then there exists one and only one non empty compact set $K\subset X$ such that $T(K)=K$.

Title self-similar fractals SelfsimilarFractals 2013-03-22 16:05:12 2013-03-22 16:05:12 paolini (1187) paolini (1187) 11 paolini (1187) Definition msc 28A80