Defining complete and observable chaos

Víctor Jiménez López

Annales Polonici Mathematici (1996)

  • Volume: 64, Issue: 2, page 139-151
  • ISSN: 0066-2216

Abstract

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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 → ∞ | f n ( x ) - f n ( y ) | = 0 and l i m s u p 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.

How to cite

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Víctor Jiménez López. "Defining complete and observable chaos." Annales Polonici Mathematici 64.2 (1996): 139-151. <http://eudml.org/doc/269968>.

@article{VíctorJiménezLópez1996,
abstract = {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.},
author = {Víctor Jiménez López},
journal = {Annales Polonici Mathematici},
keywords = {chaos in the sense of Li and Yorke; dense chaos; generic chaos; full chaos; scrambled set; topological dynamics; iteration; iterative systems; chaos},
language = {eng},
number = {2},
pages = {139-151},
title = {Defining complete and observable chaos},
url = {http://eudml.org/doc/269968},
volume = {64},
year = {1996},
}

TY - JOUR
AU - Víctor Jiménez López
TI - Defining complete and observable chaos
JO - Annales Polonici Mathematici
PY - 1996
VL - 64
IS - 2
SP - 139
EP - 151
AB - 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.
LA - eng
KW - chaos in the sense of Li and Yorke; dense chaos; generic chaos; full chaos; scrambled set; topological dynamics; iteration; iterative systems; chaos
UR - http://eudml.org/doc/269968
ER -

References

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