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1) Is there any notion of Li-Yorke chaos for non compact (metric) spaces $X$ and non continuous transformation $f:X \rightarrow X$? Could you bring some references?

2) I mean, why are so important the compactness of the space $X$ and continuity of the function $f:X \rightarrow X$ in the Li-Yorke chaos definition?

Just to remember, a pair $x,y \in X$ is called scrabled if $\liminf d(f^{n}(x),f^{n}(y)) = 0$ and $\limsup d(f^{n}(x),f^{n}(y)) > 0$ ($d$ is the metric). A set $S \subset X$ is called scrambled if every pair $x,y \in S$ is scrambled. Finally, a system $(X,d,f)$ is Li-Yorke chaotic if there is a uncountable scrambled set $S \subset X$.

Thanks for your attention

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In fact the definition works without continuity and compactness. Question is, is it good for anything? The notion originates from (continuous) interval dynamics, where it links heavily with Sharkovsky theorem and entropy. In more general spaces (still compact and with continuous map) it is implied by positive entropy but not vice-versa and in fact many examples show that the notion is in some sense artificial (some rather trivial systems reveal it). If you want a version which has nothing to do with the topological structure, there is one: measure-theoretic chaos, see https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/measure-theoretic-chaos/7802F5DC4ECEEF8A691649BA8EE78B73

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  • $\begingroup$ Dear @downar, thanks for your answer. Sometimes you can have maps which are continuous in all space but in a single or more points and the space is not compact. I believe the notion of chaos can be interesting in such context, even the Li-Yorke notion of chaos... thanks for your attention... $\endgroup$ – Bruno Brogni Uggioni Apr 30 '18 at 19:11

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