Defining complete and observable chaos

Víctor Jiménez López

Annales Polonici Mathematici, Tome 63 (1996), p. 139-151 / Harvested from The Polish Digital Mathematics Library

Résumé

For a continuous map f from a real compact interval I into itself, we consider the set C(f) of points (x,y) ∈ I² for which $l i m i n f_{n \to \infty} | f^{n} (x) - f^{n} (y) | = 0$ and $l i m s u p_{n \to \infty} | f^{n} (x) - f^{n} (y) | > 0$ . We prove that if C(f) has full Lebesgue measure then it is residual, but the converse may not hold. Also, if λ² denotes the Lebesgue measure on the square and Ch(f) is the set of points (x,y) ∈ C(f) for which neither x nor y are asymptotically periodic, we show that λ²(C(f)) > 0 need not imply λ²(Ch(f)) > 0. We use these results to propose some plausible definitions of “complete” and “observable” chaos.

Publié le : 1996-01-01

Zbl 0867.58045

EUDML-ID : urn:eudml:doc:269968

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     year = {1996},
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Víctor Jiménez López. Defining complete and observable chaos. Annales Polonici Mathematici, Tome 63 (1996) pp. 139-151. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv64z2p139bwm/

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