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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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.
M. V. Jakobson, A. M. Stepin (1989)
Banach Center Publications
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de Melo, W. (1998)
Documenta Mathematica
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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...
Michal Misiurewicz (1989)
Banach Center Publications
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Kozlovski, O.S. (2000)
Annals of Mathematics. Second Series
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Michael Yampolsky (2003)
Publications Mathématiques de l'IHÉS
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Lluis Alsedà, Jaume Llibre (1989)
Banach Center Publications
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Jerzy Dydak (1974)
Colloquium Mathematicae
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Z. Ivković, J. Vukmirović (1976)
Matematički Vesnik
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Andersen, Allan T., Barker, V.A., Nielsen, Bo Friis (1999)
Mathematical Problems in Engineering
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Andrzej Ehrenfeucht, Edward Grzegorek (1974)
Colloquium Mathematicae
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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...
Tomasz Nowicki (1993)
Fundamenta Mathematicae
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We study 1) the slopes of central branches of iterates of S-unimodal maps, comparing them to the derivatives on the critical trajectory, 2) the hyperbolic structure of Collet-Eckmann maps estimating the exponents, and under a summability condition 3) the images of the density one under the iterates of the Perron-Frobenius operator, 4) the density of the absolutely continuous invariant measure.