# self-similar fractals

Let $(X,d)$ be a metric space and let ${T}_{1},\mathrm{\dots},{T}_{N}$ be a finite number of contractions on $X$ i.e. each ${T}_{i}:X\to X$ enjoys the property

$$d({T}_{i}(x),{T}_{i}(y))\le {\lambda}_{i}d(x,y)$$ |

(${T}_{i}$ is ${\lambda}_{i}$-Lipschitz) for some $$.

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 $T\mathit{}\mathrm{(}K\mathrm{)}\mathrm{=}K$ (invariant set) is called a *self-similar fractal* with respect to the contractions $\mathrm{\{}{T}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{T}_{N}\mathrm{\}}$.

The most famous example of self-similar fractal is the Cantor set^{}.
This is constructed in $X=\mathbb{R}$ with the usual Euclidean metric^{} structure^{}, by
choosing $N=2$ contractions: ${T}_{1}(x)=x/3$, ${T}_{2}(x)=1-(1-x)/3$.

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

By choosing other appropriate transformations one can obtain the beautiful example of the Barnsley Fern^{}, which shows how the fractal^{} geometry can successfully describe nature.

An important result is given by the following Theorem^{}.

###### Theorem 1.

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

Notice that the empty set^{} always satisfies the relation^{} $T(\mathrm{\varnothing})=\mathrm{\varnothing}$
and hence is not an interesting case. On the other hand, if at least one of the ${T}_{i}$ is surjective^{} (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 |