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

Tricot, Claude; Riedi, Rudolf

Ann. Math. Blaise Pascal 6, No. 1, 55-72 (1999).

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.

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

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).