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

Fundamenta Mathematicae (2010)

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

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Abstract

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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

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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 -

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