On the ω-limit sets of tent maps

Abstract

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.