The space of $\omega $-limit sets of a continuous map of the interval

Blokh, Alexander; Bruckner, A. M.; Humke, Paul D.; Smı́tal, J.

doi:10.1090/s0002-9947-96-01600-5

Cited by 57 publications

(42 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Clearly every map which is locally eventually onto is also locally precritical, but the converse is not true in general. To illustrate these properties we recall an example from [4].…”

Section: Symbolic Dynamics and Kneading Theory Consider A Finite Al-mentioning

confidence: 99%

“…The asymptotic behaviour of f at the point x can be ascertained by studying the structure of the ω-limit set at that point. The structure of ω-limit sets for maps on a compact interval has been investigated by many authors, including Block and Coppel [3] and Hirsch et al [9], and characterizations of such sets have been given by Balibrea and La Paz [1], Blokh et al [4] and by this author with Good, Knight and Raines [2]. In this paper we give a characterization based on the metric property of internal chain transitivity.…”

mentioning

confidence: 97%

“…The characterizations of ω-limit sets for interval maps found in [1] and [4] contain definitions relating specifically to interval maps; in this paper we rely largely upon symbolic dynamics for our analysis, which allows us to give our characterization requiring internal chain transitivity only. Furthermore, we extend a well-known test using the limit-itineraries of points in the interval to check whether any point is in the ω-limit set of any other point (Theorem 5.4).…”

mentioning

confidence: 99%

“…Remark 4. 4. It may appear that the condition of Λ not containing the image of any critical point is simply an artefact of using symbolic dynamics.…”

mentioning

confidence: 99%

“…We call the element s(Λ, N ) in the above construction an arbitrary length, infinite repetition sequence for the internally chain transitive set Λ.Lemma 3 4…”

mentioning

confidence: 99%

See 4 more Smart Citations

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

Barwell¹

2010

Fund. Math.

View full text Add to dashboard Cite

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.

show abstract

“…Clearly every map which is locally eventually onto is also locally precritical, but the converse is not true in general. To illustrate these properties we recall an example from [4].…”

Section: Symbolic Dynamics and Kneading Theory Consider A Finite Al-mentioning

confidence: 99%

mentioning

confidence: 97%

mentioning

confidence: 99%

“…Remark 4. 4. It may appear that the condition of Λ not containing the image of any critical point is simply an artefact of using symbolic dynamics.…”

mentioning

confidence: 99%

“…We call the element s(Λ, N ) in the above construction an arbitrary length, infinite repetition sequence for the internally chain transitive set Λ.Lemma 3 4…”

mentioning

confidence: 99%

See 3 more Smart Citations