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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm207-2-4,
author = {Andrew D. Barwell},
title = {A characterization of $\omega$-limit sets for piecewise monotone maps of the interval},
journal = {Fundamenta Mathematicae},
volume = {209},
year = {2010},
pages = {161-174},
zbl = {1201.37014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm207-2-4}
}
Andrew D. Barwell. A characterization of ω-limit sets for piecewise monotone maps of the interval. Fundamenta Mathematicae, Tome 209 (2010) pp. 161-174. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm207-2-4/