Defining complete and observable chaos
Annales Polonici Mathematici, Volume 64 (1996) no. 2, pp. 139-151
See the original article notice from the Institute of Mathematics Polish Academy of Sciences source
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.
Keywords:
chaos in the sense of Li and Yorke, dense chaos, generic chaos, full chaos, scrambled set
Víctor López. Defining complete and observable chaos. Annales Polonici Mathematici, Volume 64 (1996) no. 2, pp. 139-151. doi: 10.4064/ap-64-2-139-151
@article{10_4064_ap_64_2_139_151,
author = {V{\'\i}ctor L\'opez},
title = {Defining complete and observable chaos},
journal = {Annales Polonici Mathematici},
pages = {139--151},
year = {1996},
volume = {64},
number = {2},
doi = {10.4064/ap-64-2-139-151},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/ap-64-2-139-151/}
}
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