A characterization of ω-limit sets for piecewise monotone maps of the interval

Andrew D. Barwell

Fundamenta Mathematicae (2010)

  • Volume: 207, Issue: 2, page 161-174
  • ISSN: 0016-2736

Abstract

top
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.

How to cite

top

Andrew D. Barwell. "A characterization of ω-limit sets for piecewise monotone maps of the interval." Fundamenta Mathematicae 207.2 (2010): 161-174. <http://eudml.org/doc/283314>.

@article{AndrewD2010,
abstract = {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.},
author = {Andrew D. Barwell},
journal = {Fundamenta Mathematicae},
keywords = {omega-limit set; piecewise monotone map; symbolic dynamics; kneading theory},
language = {eng},
number = {2},
pages = {161-174},
title = {A characterization of ω-limit sets for piecewise monotone maps of the interval},
url = {http://eudml.org/doc/283314},
volume = {207},
year = {2010},
}

TY - JOUR
AU - Andrew D. Barwell
TI - A characterization of ω-limit sets for piecewise monotone maps of the interval
JO - Fundamenta Mathematicae
PY - 2010
VL - 207
IS - 2
SP - 161
EP - 174
AB - 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.
LA - eng
KW - omega-limit set; piecewise monotone map; symbolic dynamics; kneading theory
UR - http://eudml.org/doc/283314
ER -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.