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Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos. (English) Zbl 0997.54061

Let $X$ 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.

##### MSC:
 54H20 Topological dynamics 37B10 Symbolic dynamics 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE)