A characterization of $\omega$-limit sets for piecewise monotone maps of the interval
Volume 207 / 2010
Abstract
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.
