( is -Lipschitz) for some .
Given a set we can define
A set such that (invariant set) is called a self-similar fractal with respect to the contractions .
A more interesting example is the Koch curve in . In this case we choose similitudes with factor .
An important result is given by the following Theorem.
Notice that the empty set always satisfies the relation and hence is not an interesting case. On the other hand, if at least one of the is surjective (as happens in the examples above), then the whole set satisfies .