For a continuous map f on a compact metric space (X,d), a subset D of X is
internally chain transitive if for every x and y in D and every delta > 0 there
is a sequence of points {x=x_0,x_1, ...,x_n=y} such that d(f(x_i),x_{i+1}) <
delta for i=0,1, ...,n-1. 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 subset D of 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
tent map with slope equal to 2. In this paper, we prove that for tent maps
whose critical point c=1/2 is periodic, 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 those
tent maps with shadowing (or pseudo-orbit tracing), the omega-limit sets are
precisely those sets having internal chain transitivity.Comment: 17 page