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Distortion results and invariant Cantor sets of unimodal maps. (English) Zbl 0809.58026
The author proves that Cantor attractors of $$S$$-unimodal maps have Lebesgue measure zero. A distortion theory is developed for these maps. It is proved, that every $$S$$-unimodal map not having periodic attractors has weak-Markov property (this means, that the map has uniform good distortion properties).

##### MSC:
 37A99 Ergodic theory 37D99 Dynamical systems with hyperbolic behavior
##### Keywords:
$$S$$-unimodal maps; distortion theory
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##### References:
 [1] DOI: 10.2307/1971501 · Zbl 0708.58007 [2] DOI: 10.1007/BF01205554 · Zbl 0625.58027 [3] Blokh, Measurable dynamics of S-unimodal maps of the interval (1990) · Zbl 0790.58024 [4] DOI: 10.1007/BF01077983 · Zbl 0623.14026 [5] Hofbauer, Ergod. Th. & Dynam. Sys. 1 pp 159– (1981) [6] Misiurewicz, Publ. Math. IHES 53 pp 17– (1981) · Zbl 0477.58020 [7] DOI: 10.1007/BF02392981 · Zbl 0761.58007 [8] Keller, Ergod. Th. & Dynam. Sys. 10 pp 717– (1990) [9] Jacobson, Metric properties of non-renormalizable (1991) [10] DOI: 10.1007/BF01212280 · Zbl 0595.58028
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