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**Attractors, orbits and ergodicity.
(Attracteurs, orbites et ergodicité.)**
*(French)*
Zbl 1054.37014

Summary: The attractor of a system of iterated functions is the support of a measure called invariant or auto-similar: it is the fixed point of the Markov operator in the metric space of probability measures with compact support. A random algorithm permits the construction of the attractor which is almost sure the limit points set of an orbit. With the help of an ergodic theorem one can prove that the visit frequency of a set through this orbit is almost sure equal to the invariant measure of this set.

We give a simple proof of this known result.

We give a simple proof of this known result.

### MSC:

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

28D05 | Measure-preserving transformations |

37A25 | Ergodicity, mixing, rates of mixing |

37B10 | Symbolic dynamics |

37C45 | Dimension theory of smooth dynamical systems |

28A80 | Fractals |

### Keywords:

iterated function system; invariant measure; attractor; ergodic theorem; fractals; orbits; Cantor sets
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\textit{C. Tricot} and \textit{R. Riedi}, Ann. Math. Blaise Pascal 6, No. 1, 55--72 (1999; Zbl 1054.37014)

### References:

[1] | Barnsley, M., Fractals Everywhere, Academic Press (1988). · Zbl 0691.58001 |

[2] | Elton, J., An Ergodic Theorem for Iterated Maps, Journal of Ergodic theory and Dynamical Systems, 7 (1987), 481-488. · Zbl 0621.60039 |

[3] | Hutchinson, J., Fractals and Self-similarity, Indiana University Journal of Mathematics, 30 (1981), 713-747. · Zbl 0598.28011 |

[4] | Peitgen, H.-O., Chaos and Fractals, Springer-Verlag (1990). |

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