For a piecewise monotone map f on a compact interval I, we characterize the ω-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 Λ⊂I is closed, invariant and contains no post-critical point, then Λ is the ω-limit set of a point in I if and only if Λ is internally chain transitive in the sense of Hirsch, Smith and Zhao; the proof relies upon symbolic dynamics. By identifying points of ω-limit sets via their limit-itineraries, we offer simple examples which show that internal chain transitivity does not characterize ω-limit sets for interval maps in general.
|Translated title of the contribution||A characterization of ω-limit sets for piecewise monotone maps of the interval|
|Pages (from-to)||161 - 174|
|Number of pages||14|
|Volume||207, issue 2|
|Publication status||Published - Apr 2010|