Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos.

*(English)*Zbl 0997.54061Let \(X\) be a compact metric space, \(f:X\to X\) be a continuous surjection. The authors consider the dynamical system induced on \(X\) by \(f\) and discuss relations between definitions of chaos introduced by Devaney and Li-Yorke as well as the notion of scattering and 2-scattering introduced by Blanchard, Host and Maas. In particular, it is proved that chaos in the sense of Devaney is stronger than that of Li and Yorke.

In Sect. 1 (Introduction) necessary definitions are recalled (introduced) and some introductory propositions are proved. In particular, the notion of sensitivity, transitivity and total transitivity of \(f\) is defined and sensitivity is characterized by means of almost equicontinuity. Sect. 2 contains results on asymptotic relations. Proximal relation and scrambled sets are discussed in Sect. 3. Some applications of results proved in Sections 2 and 3 are given in Sect. 4. Theorem 4.1 says that if \(f\) is transitive (with \(X\) infinite) and there is a periodic point, then there is an uncountable scrambled set for \(f\). Moreover, if \(f\) is totally transitive, then \(f\) is densely chaotic in the sense of Li and Yorke. Thus, chaos in the sense of Devaney is stronger than that in the sense of Li and Yorke. From Theorem 4.3 it follows in particular that if \(f\) is 2-scattering then \(f\) has a dense, uncountable, scrambled set. In the ‘Appendix’ it is proved that some properties hold for homeomorphisms are true for continuous maps.

In Sect. 1 (Introduction) necessary definitions are recalled (introduced) and some introductory propositions are proved. In particular, the notion of sensitivity, transitivity and total transitivity of \(f\) is defined and sensitivity is characterized by means of almost equicontinuity. Sect. 2 contains results on asymptotic relations. Proximal relation and scrambled sets are discussed in Sect. 3. Some applications of results proved in Sections 2 and 3 are given in Sect. 4. Theorem 4.1 says that if \(f\) is transitive (with \(X\) infinite) and there is a periodic point, then there is an uncountable scrambled set for \(f\). Moreover, if \(f\) is totally transitive, then \(f\) is densely chaotic in the sense of Li and Yorke. Thus, chaos in the sense of Devaney is stronger than that in the sense of Li and Yorke. From Theorem 4.3 it follows in particular that if \(f\) is 2-scattering then \(f\) has a dense, uncountable, scrambled set. In the ‘Appendix’ it is proved that some properties hold for homeomorphisms are true for continuous maps.

Reviewer: Andrzej Pelczar (Kraków)

##### MSC:

54H20 | Topological dynamics (MSC2010) |

37B10 | Symbolic dynamics |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

##### Keywords:

Devaney’s chaos; Li-Yorke chaos; proximal relations; scattering; sensitivity; transitivity; asymptotic relations; scrambled set
PDF
BibTeX
XML
Cite

\textit{W. Huang} and \textit{X. Ye}, Topology Appl. 117, No. 3, 259--272 (2002; Zbl 0997.54061)

Full Text:
DOI

##### References:

[1] | Akin, E., The general topology of dynamical systems, (1993), American Mathematical Society Providence, RI · Zbl 0781.54025 |

[2] | Auslander, J., Minimal flows and their extensions, North-holland math. stud., 153, (1988), North-Holland Amsterdam · Zbl 0654.54027 |

[3] | E. Akin, J. Auslander, E. Glasner, Residual properties and almost equicontinuity, Preprint, 1999 · Zbl 1182.37009 |

[4] | Akin, E.; Auslander, J.; Berg, K., When is a transitive map chaotic?, () · Zbl 0861.54034 |

[5] | Auslander, J.; Yorke, J.A., Interval maps, factors of maps and chaos, Tohoku math. J., 32, 177-188, (1980) · Zbl 0448.54040 |

[6] | Banks, J.; Brooks, J.; Cairns, G.; Davis, G.; Stacey, P., On Devaney’s definition of chaos, Amer. math. monthly, 99, 332-334, (1992) · Zbl 0758.58019 |

[7] | F. Blanchard, B. Host, S. Ruette, Asymptotic pairs in positive-entropy systems, Preprint, 2000 · Zbl 1018.37005 |

[8] | Blanchard, F.; Host, B.; Maass, A., Topological complexity, Ergodic theory dynamical systems, 20, 641-662, (2000) · Zbl 0962.37003 |

[9] | F. Blanchard, E. Glasner, S. Kolyada, A. Maass, On Li-Yorke pairs, Preprint, 2000 |

[10] | Devaney, R., Chaotic dynamical systems, (1989), Addison-Wesley Reading MA |

[11] | Furstenberg, H., Recurrence in ergodic theory and combinatorial number theory, (1981), Princeton · Zbl 0459.28023 |

[12] | Glasner, E., A simple characterization of the set of μ-entropy pairs and applications, Israel J. math., 102, 13-27, (1997) · Zbl 0909.54035 |

[13] | W. Huang, X. Ye, Homeomorphisms with the whole compacta being scrambled sets, in: Ergodic Theory Dynamical Systems, to appear · Zbl 0978.37003 |

[14] | Glasner, E.; Weiss, B., Sensitive dependence on initial conditions, Nonlinearity, 6, 1067-1075, (1993) · Zbl 0790.58025 |

[15] | Iwanik, A., Independent sets of transitive points, (), 277-282 · Zbl 0703.54024 |

[16] | Li, T.; Yorke, J., Period 3 implies chaos, Amer. math. monthly, 82, 985-992, (1975) · Zbl 0351.92021 |

[17] | J.Xiong, Z. Yang, Chaos caused by a topologically mixing map, in: World Sci. Adv. Ser. Dyn. Syst., Vol. 9, pp. 550-572 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.