Attractors of maps of the interval
A. M. Blokh, M. Yu. Lyubich (1989)
Banach Center Publications
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A. M. Blokh, M. Yu. Lyubich (1989)
Banach Center Publications
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Michal Misiurewicz (1989)
Banach Center Publications
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Krystyna Ziemian (1989)
Studia Mathematica
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Andrew D. Barwell (2010)
Fundamenta Mathematicae
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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...
Kozlovski, O.S. (2000)
Annals of Mathematics. Second Series
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Jeyakumar, V., Luc, D.T., Schaible, S. (1998)
Journal of Convex Analysis
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Lluis Alsedà, Jaume Llibre (1989)
Banach Center Publications
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Simin Li, Weixiao Shen (2006)
Fundamenta Mathematicae
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It is proved that a smooth unimodal interval map with critical order 2 + ε has no wild attractor if ε >0 is small.
de Melo, W. (1998)
Documenta Mathematica
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F. Balibrea, C. La Paz (1997)
Annales Polonici Mathematici
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One-dimensional turbulent maps can be characterized via their ω-limit sets [1]. We give a direct proof of this characterization and get stronger results, which allows us to obtain some other results on ω-limit sets, which previously were difficult to prove.
Andrew D. Barwell, Chris Good, Piotr Oprocha (2012)
Fundamenta Mathematicae
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We address various notions of shadowing and expansivity for continuous maps restricted to a proper subset of their domain. We prove new equivalences of shadowing and expansive properties, we demonstrate under what conditions certain expanding maps have shadowing, and generalize some known results in this area. We also investigate the impact of our theory on maps of the interval.
Diogo Baptista, Ricardo Severino (2012)
ESAIM: Proceedings
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Building on the kneading theory for Lozi maps introduced by Yutaka Ishii, in 1997, we introduce a symbolic method to compute its largest Lyapunov exponent. We use this method to study the behavior of the largest Lyapunov exponent for the set of points whose forward and backward orbits remain bounded, and find the maximum value that the largest Lyapunov exponent can assume.
Gregory J. Davis (1990)
Publicacions Matemàtiques
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In this paper we provide a direct proof of hyperbolicity for a class of one-dimensional maps on the unit interval. The maps studied are degenerate forms of the standard quadratic map on the interval. These maps are important in understanding the Newhouse theory of infinitely many sinks due to homoclinic tangencies in two dimensions.
Andrzej Ehrenfeucht, Edward Grzegorek (1974)
Colloquium Mathematicae
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Jerzy Dydak (1974)
Colloquium Mathematicae
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Władysław A. Majewski, Marcin Marciniak (2007)
Banach Center Publications
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The structure of the set of positive unital maps between M₂(ℂ) and Mₙ(ℂ) (n ≥ 3) is investigated. We proceed with the study of the "quantized" Choi matrix thus extending the methods of our previous paper [MM2]. In particular, we examine the quantized version of Størmer's extremality condition. Maps fulfilling this condition are characterized. To illustrate our approach, a careful analysis of Tang's maps is given.
Magnus Aspenberg (2009)
Fundamenta Mathematicae
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We show that Misiurewicz maps for which the Julia set is not the whole sphere are Lebesgue density points of hyperbolic maps.