Topics from One-Dimensional Dynamics

Brucks, Karen M.; Bruin, Henk

doi:10.1017/cbo9780511617171

Cited by 67 publications

(83 citation statements)

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“…It is carefully written and is useful both for the beginner and the experienced researcher. The book by Brucks and Bruin [15] covers a wide range of topics from one-dimensional dynamics. It would be a good text for students preparing for research in the area.…”

Section: Further Readingmentioning

confidence: 99%

Attractors and inverse limits

Keesling

2008

Rev. R. Acad. Cien. Serie A. Mat.

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This paper surveys some recent results concerning inverse limits of tent maps. The survey concentrates on Ingram's Conjecture. Some motivation is given for the study of such inverse limits. Atractores y límites inversosResumen. Este artículo expone algunos resultados recientes sobre límites inversos de aplicaciones tienda. La exposición se concentra en la Conjetura de Ingram. Se presentan tambien algunas motivaciones para el estudio de tales límites inversos.

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Section: Further Readingmentioning

confidence: 99%

Attractors and inverse limits

Keesling

2008

Rev. R. Acad. Cien. Serie A. Mat.

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“…In many recent texts, tent maps are cited as examples of simple maps with complicated and interesting dynamics [3,13,16,19]. Furthermore, they have been the subject of many research articles [12,17,28,30], often in relation to their ω-limit sets [6,21,22,23,24].…”

Section: Introductionmentioning

confidence: 99%

On the ω-limit sets of tent maps

2012

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Abstract. For a continuous map f on a compact metric space (X, d), a set D ⊂ X is internally chain transitive if for every x, y ∈ D and every δ > 0 there is a sequence of points x = x 0 , x 1 , . . . , x n = y such that d(f (x i ), x i+1 ) < δ for 0 ≤ i < n. It is known that every ω-limit set is internally chain transitive; in earlier work it was shown that for X a shift of finite type, a closed set D ⊂ X is internally chain transitive if and only if D is an ω-limit set for some point in X, and that the same is also true for the full tent map T 2 : [0, 1] → [0, 1]. In this paper, we prove that for tent maps with periodic critical point every closed, internally chain transitive set is necessarily an ω-limit set. Furthermore, we show that there are at least countably many tent maps with non-recurrent critical point for which there is a closed, internally chain transitive set which is not an ω-limit set. Together, these results lead us to conjecture that for maps with shadowing, the ω-limit sets are precisely those sets having internal chain transitivity.

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“…Such a map is called a monotonic mod one transformation (with two monotonic pieces). A general monotonic mod one transformation is also defined as in (1), but f : [0, 1] → R. Assume that f is a piecewise differentiable function, which means f is differentiable on (0, 1) \ F where F is a finite set. The map T f is called topologically transitive if there is an x ∈ [0, 1] such that {T f n x : n ∈ N} is dense in [0,1].…”

Section: Introductionmentioning

confidence: 99%