Devaneyʹs chaos or 2-scattering implies Li–Yorkeʹs chaos

Let X be a compact metric space, and let f :X→X be transitive with X infinite. We show that each asymptotic class (or the stable set Ws(x) for each x∈X) is of first category and so is the asymptotic relation. Moreover, we prove that if the proximal relation is dense in a neighbourhood of some point in the diagonal then f is chaotic in the sense of Li–Yorke. As applications we obtain that if f contains a periodic point, or f is 2-scattering, then f is chaotic in the sense of Li–Yorke. Thus, chaos in the sense of Devaney is stronger than that of Li–Yorke.