Non-uniformly expanding dynamics in maps with singularities and criticalities
Stephano Luzzatto, Warwick Tucker (1999)
Publications Mathématiques de l'IHÉS
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Stephano Luzzatto, Warwick Tucker (1999)
Publications Mathématiques de l'IHÉS
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Anthony Quas (1999)
Studia Mathematica
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We consider the topological category of various subsets of the set of expanding maps from a manifold to itself, and show in particular that a generic expanding map of the circle has no absolutely continuous invariant probability measure. This is in contrast with the situation for or expanding maps, for which it is known that there is always a unique absolutely continuous invariant probability measure.
Tomasz Nowicki, Sebastian Van Strien (1990)
Annales de l'I.H.P. Physique théorique
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T. Nowicki, S. van Strien (1988)
Inventiones mathematicae
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B. Schmitt (1989)
Banach Center Publications
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A. M. Blokh, M. Yu. Lyubich (1991)
Annales scientifiques de l'École Normale Supérieure
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John N. Mather (1986)
Publications Mathématiques de l'IHÉS
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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.
Kozlovski, O.S. (2000)
Annals of Mathematics. Second Series
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