A note on generic chaos

• Autorzy: Gongfu Liao
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

metric space; dynamical system; topological mixing; generic chaos

Kategorie tematyczne

• 54H20: Topological dynamics
• 70K50: Bifurcations and instability

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1994
• Tom: 59
• Numer: 2
• Strony: 99-105

Daty

wydano

1994

otrzymano

1993-01-04

poprawiono

1993-04-20

Twórcy

autor

Gongfu Liao

• Department of Mathematics, Jilin University, Changchun, Jilin, People's Republic of China

Bibliografia

• [1] L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer, 1992.
• [2] W. A. Coppel, Chaos in one dimension, in: Chaos and Order (Canberra, 1990), World Sci., Singapore, 1991, 14-21.
• [3] K. Janková and J. Smítal, A characterization of chaos, Bull. Austral. Math. Soc. 34 (1986), 283-292.
• [4] T.-Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985-992.
• [5] G.-F. Liao, ω-limit sets and chaos for maps of the interval, Northeastern Math. J. 6 (1990), 127-135.
• [6] M. Osikawa and Y. Oono, Chaos in C⁰-endomorphism of interval, Publ. Res. Inst. Math. Sci. 17 (1981), 165-177.
• [7] K. Petersen, Ergodic Theory, Cambridge University Press, 1983.
• [8] J. Piórek, On the generic chaos in dynamical systems, Univ. Iagell. Acta Math. 25 (1985), 293-298.
• [9] J. Piórek, On generic chaos of shifts in function spaces, Ann. Polon. Math. 52 (1990), 139-146.