Displaying similar documents to “Attractors of maps of the interval”

On unimodal maps with critical order 2 + ε

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.

Higher order Schwarzian derivatives in interval dynamics

O. Kozlovski, D. Sands (2009)

Fundamenta Mathematicae

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We introduce an infinite sequence of higher order Schwarzian derivatives closely related to the theory of monotone matrix functions. We generalize the classical Koebe lemma to maps with positive Schwarzian derivatives up to some order, obtaining control over derivatives of high order. For a large class of multimodal interval maps we show that all inverse branches of first return maps to sufficiently small neighbourhoods of critical values have their higher order Schwarzian derivatives...

Shadowing and expansivity in subspaces

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.

Hyperbolicity in a class of one-dimensional maps.

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.

Turbulent maps and their ω-limit sets

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.

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

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...