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Li–Yorke chaos in higher dimensions: a review. (English) Zbl 1096.39019

This paper presents a review on the topic of Li-Yorke chaos for discrete dynamical systems $\left(X,\varphi \right),$ where $X$ is a metric space and $\varphi$ is meant a continuous map from $X$ into itself. The survey starts with the notion of chaos in the sense of Li and Yorke, and with their famous condition “period three implies chaos” for providing it, in the case of unidimensional maps [see T.-Y. Li and J. A. Yorke, Am. Math. Mon. 82, 985–992 (1975; Zbl 0351.92021)]. Next, the authors pay attention in the Marotto’s theorem [see F. R. Marotto, J. Math. Anal. Appl. 63, 199–223 (1978; Zbl 0381.58004)] as a natural extension of Li-Yorke’s chaos to differentiable maps $\varphi :{ℝ}^{n}\to {ℝ}^{n}$ having snap-back repeller fixed points. This result was in turn generalized by P. Kloeden [J. Aust. Math. Soc. Ser. A 31, 217–225 (1981; Zbl 0471.39001)] under the assumption of continuity for $\varphi$, and it was allowed to $\varphi$ to have saddle-points.

The paper under review shows a proof of this last result containing at the same time the original proofs of Li and Yorke and that of Marotto. Moreover, the authors provide two examples in order to illustrate it, one of them (the tent map) having snap-back repellers, and the other one (a twisted horseshoe of triangular type on the unit square ${\left[0,1\right]}^{2}$) possessing a saddle point. The survey continues by showing the statements of some extensions of the before mentioned result of Kloeden to Banach spaces and to metric spaces of fuzzy sets. Finally, the authors review some results on chaotic maps in the sense of Devaney, in the setting of complete metric spaces $X$. The key for obtaining them is to find a Cantor set ${\Lambda }\subset X$ such that $\varphi :{\Lambda }\to {\Lambda }$ is topologically conjugated to the (chaotic) discrete dynamical system generated by the shift map $\sigma$ on the space ${{\Sigma }}_{2}$ of sequences of 0’s and 1’s.

##### MSC:
 39A12 Discrete version of topics in analysis 37D45 Strange attractors, chaotic dynamics