On the omega-limit sets of tent maps

For a continuous map f on a compact metric space (X, d), a set D ⊂ X is internally chain transitive if for every x, y ∈ D and every > 0 there is a sequence of points hx = x0, x1, . . . , xn = yi such that d(f(xi), xi+1) < for 0 ≤ i <n. It is known that every !-limit set is internally chain transitive; in earlier work it was shown that for X a shift of finite type, a closed set D ⊂ X is internally chain transitive if and only if D is an !-limit set for some point in X, and that the same is also true for the full tent map T2 : [0, 1] → [0,1]. In this paper, we prove that for tent maps with periodic critical point every closed, internally chain transitive set is necessarily an !-limit set. Furthermore, we show that there are at least countably many tent maps with non-recurrent critical point for which there is a closed, internally chain transitive set which is not an !-limit set. Together, these results lead us to conjecture that for those tent maps with shadowing, the !-limit sets are precisely those sets having internal chain transitivity.