On the ω-limit sets of tent maps

For a continuous map \(f\) on a compact metric space $(X,d)$, a set D C X is \(internally\) \(chain\) \(transitive\) if for every x,y \(\epsilon\) D and every \(\sigma\)0> O there is a sequence of points x=x\(_0\),x\(_1\),.....,x\(_n\)=y such that d(f(x\(_i\)) \(_,\)x\(_i\)\(_+\)\(_1\))<\(\sigma\)for 0\(\leq\) i < n. It is known that every \(\omega\)-limit set is internally chain transitive; in earlier work it was shown that for X a shift of finite type, a closed set D C X is internally chain transitive if and only if D is an \(\omega\)-limit set for some point in X, and that the same is also true for the full tent map T\(_2\):[0,1]\(\rightarrow\)[0,1]. In this paper, we prove that for tent maps with periodic critical point every closed, internally chain transitive set is necessarily an \(\omega\)-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 \(\omega\)-limit set. Together, these results lead us to conjecture that for maps with \(shadowing\), the \(\omega\)-limit sets are precisely those sets having internal chain transitivity.