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