We consider dynamical systems on a separable metric space containing at least two points. It is proved that weak topological mixing implies generic chaos, but the converse is false. As an application, some results of Piórek are simply reproved.
@article{bwmeta1.element.bwnjournal-article-apmv59z2p99bwm,
author = {Gongfu Liao},
title = {A note on generic chaos},
journal = {Annales Polonici Mathematici},
volume = {60},
year = {1994},
pages = {99-105},
zbl = {0810.54032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv59z2p99bwm}
}
Gongfu Liao. A note on generic chaos. Annales Polonici Mathematici, Tome 60 (1994) pp. 99-105. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv59z2p99bwm/
[000] [1] L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer, 1992.
[001] [2] W. A. Coppel, Chaos in one dimension, in: Chaos and Order (Canberra, 1990), World Sci., Singapore, 1991, 14-21.
[002] [3] K. Janková and J. Smítal, A characterization of chaos, Bull. Austral. Math. Soc. 34 (1986), 283-292. | Zbl 0577.54041
[003] [4] T.-Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985-992. | Zbl 0351.92021
[004] [5] G.-F. Liao, ω-limit sets and chaos for maps of the interval, Northeastern Math. J. 6 (1990), 127-135.
[005] [6] M. Osikawa and Y. Oono, Chaos in C⁰-endomorphism of interval, Publ. Res. Inst. Math. Sci. 17 (1981), 165-177. | Zbl 0468.58012
[006] [7] K. Petersen, Ergodic Theory, Cambridge University Press, 1983.
[007] [8] J. Piórek, On the generic chaos in dynamical systems, Univ. Iagell. Acta Math. 25 (1985), 293-298. | Zbl 0587.54061
[008] [9] J. Piórek, On generic chaos of shifts in function spaces, Ann. Polon. Math. 52 (1990), 139-146. | Zbl 0719.58005