Geodesic
Defining complete and observable chaos
Annales Polonici Mathematici, Tome 64 (1996) no. 2, pp. 139-151

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

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 $lim inf_{n→∞} |f^n(x) - f^n(y)| = 0$ and $lim sup_{n→∞} |f^n(x) - f^n(y)| > 0$. 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.
DOI : 10.4064/ap-64-2-139-151
Keywords: chaos in the sense of Li and Yorke, dense chaos, generic chaos, full chaos, scrambled set

Víctor López 1

1 
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Víctor López. Defining complete and observable chaos. Annales Polonici Mathematici, Tome 64 (1996) no. 2, pp. 139-151. doi: 10.4064/ap-64-2-139-151

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