self-similar fractals

Let (X,d) be a metric space and let T1,,TN be a finite number of contractions on X i.e. each Ti:XX enjoys the property


(Ti is λi-Lipschitz) for some λi<1.

Given a set AX we can define

Definition 1.

A set K such that T(K)=K (invariant set) is called a self-similar fractal with respect to the contractions {T1,,TN}.

The most famous example of self-similar fractal is the Cantor setMathworldPlanetmath. This is constructed in X= with the usual Euclidean metricMathworldPlanetmath structureMathworldPlanetmath, by choosing N=2 contractions: T1(x)=x/3, T2(x)=1-(1-x)/3.

A more interesting example is the Koch curveMathworldPlanetmath in X=2. In this case we choose N=4 similitudes with factor 1/3.

By choosing other appropriate transformations one can obtain the beautiful example of the Barnsley FernMathworldPlanetmath, which shows how the fractalMathworldPlanetmath geometry can successfully describe nature.

An important result is given by the following TheoremMathworldPlanetmath.

Theorem 1.

Let X be a complete metric space and let T1,,TN:XX be a given set of contractions. Then there exists one and only one non empty compact set KX such that T(K)=K.

Notice that the empty setMathworldPlanetmath always satisfies the relationMathworldPlanetmath T()= and hence is not an interesting case. On the other hand, if at least one of the Ti is surjectivePlanetmathPlanetmath (as happens in the examples above), then the whole set X satisfies T(X)=X.

Title self-similar fractals
Canonical name SelfsimilarFractals
Date of creation 2013-03-22 16:05:12
Last modified on 2013-03-22 16:05:12
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 11
Author paolini (1187)
Entry type Definition
Classification msc 28A80