# proof of existence and unicity of self-similar fractals

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

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

We claim that $T$ is a contraction. In fact, recalling that $\delta ({A}_{1}\cup {A}_{2},{B}_{1}\cup {B}_{2})\le \mathrm{max}\{\delta ({A}_{1},{B}_{1}),\delta ({A}_{2},{B}_{2})\}$ while $\delta ({T}_{i}(A),{T}_{i}(B))\le {\lambda}_{i}\delta (A,B)$ if ${T}_{i}$ is ${\lambda}_{i}$-Lipschitz^{}, we have

$\delta (T(A),T(B))$ | $=\delta ({\displaystyle \bigcup _{i}}{T}_{i}(A),{\displaystyle \bigcup _{i}}{T}_{i}(B))\le \underset{i}{\mathrm{max}}\delta ({T}_{i}(A),{T}_{i}(B))$ | ||

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

with $$.

So $T$ is a contraction on the complete metric space $\mathcal{K}(X)$ and hence,
by Banach Fixed Point Theorem^{}, there exists one and only one $K\in \mathcal{K}(X)$ such that $T(K)=K$.

Title | proof of existence and unicity of self-similar fractals |
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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 |