A characterization of omega-limit sets for piecewise monotone maps of the interval

For a piecewise monotone map f on a compact interval I, we characterize the omega-limit sets that are bounded away from the post-critical points of f. If the pre-critical points of f are dense, for example when f is locally eventually onto, and Lambda subset of I is closed, invariant and contains no post-critical point, then Lambda is the omega-limit set of a point in I if and only if Lambda is internally chain transitive in the sense of Hirsch, Smith and Zhao; the proof relies upon symbolic dynamics. By identifying points of omega-limit sets via their limit-itineraries, we offer simple examples which show that internal chain transitivity does not characterize omega-limit sets for interval maps in general.