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

Andrew D. Barwell¹

¹ School of Mathematics and Statistics University of Birmingham Birmingham, B15 2TT, UK

Fundamenta Mathematicae, Tome 207 (2010) no. 2, pp. 161-174.

Résumé

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.

Détail

DOI : 10.4064/fm207-2-4

Keywords: piecewise monotone map compact interval characterize omega limit sets bounded away post critical points pre critical points dense example locally eventually mit lambda subset closed invariant contains post critical point mit lambda omega limit set point only mit lambda internally chain transitive sense hirsch smith zhao proof relies symbolic dynamics identifying points omega limit sets via their limit itineraries offer simple examples which internal chain transitivity does characterize omega limit sets interval maps general