### 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 ${C}^{1}$ expanding map of the circle has no absolutely continuous invariant probability measure. This is in contrast with the situation for ${C}^{2}$ or ${C}^{1+\epsilon}$ 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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