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

Volume 207 / 2010

Andrew D. Barwell Fundamenta Mathematicae 207 (2010), 161-174 MSC: 37B10, 37E05, 54C05, 54H20. DOI: 10.4064/fm207-2-4


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 ${\mit\Lambda}\subset I$ is closed, invariant and contains no post-critical point, then ${\mit\Lambda}$ is the $\omega$-limit set of a point in $I$ if and only if ${\mit\Lambda}$ 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.


  • Andrew D. BarwellSchool of Mathematics and Statistics
    University of Birmingham
    Birmingham, B15 2TT, UK

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