Misiurewicz, Michal Absolutely continuous measures for certain maps of an interval. (English) Zbl 0477.58020 Publ. Math., Inst. Hautes Étud. Sci. 53, 17-51 (1981). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 13 ReviewsCited in 104 Documents MSC: 37A99 Ergodic theory 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 28D20 Entropy and other invariants 54C70 Entropy in general topology Keywords:piecewise smooth map of an interval into itself with negative Schwarzian derivative and polynomial behaviour in the neighbourhoods of exceptional points; no periodic attracting points; existence of a probability invariant measure absolutely continuous with respect to the Lebesgue measure; metric entropies; topological entropy PDF BibTeX XML Cite \textit{M. Misiurewicz}, Publ. Math., Inst. Hautes Étud. Sci. 53, 17--51 (1981; Zbl 0477.58020) Full Text: DOI Numdam EuDML References: [1] N. Bourbaki,Fonctions d’une variable réelle (Livre IV), Paris, Hermann, 1958 (chap. 1, § 4, exerc. 1a). [2] J. Guckenheimer,Sensitive dependence to initial conditions for one dimensional maps, preprint I.H.E.S. (1979). · Zbl 0429.58012 [3] M. Jakobson, Topological and metric properties of one-dimensional endomorphisms,Dokl. Akad. Nauk SSSR,243 (1978), 866–869 (in Russian). [4] J. Milnor, W. Thurston,On iterated maps of the interval, preprint. · Zbl 0664.58015 [5] M. Misiurewicz, Structure of mappings of an interval with zero entropy,Publ. Math. I.H.E.S.,53 (1981), 000-000. [6] M. Misiurewicz, W. Szlenk, Entropy of piecewise monotone mappings,Astérisque,50 (1977), 299–310 (full version will appear inStudia Math.,67). · Zbl 0376.54019 [7] W. Parry,Entropy and generators in ergodic theory, New York, Benjamin, 1969. · Zbl 0175.34001 [8] W. Parry, Symbolic dynamics and transformations of the unit interval,Trans. Amer. Math. Soc.,122 (1966), 368–378. · Zbl 0146.18604 · doi:10.1090/S0002-9947-1966-0197683-5 [9] R. Shaw,Strange attractors, chaotic behavior and information flow, preprint, Santa Cruz (1978). [10] D. Singer, Stable orbits and bifurcation of maps of the interval,SIAM J. Appl. Math.,35 (1978), 260–267. · Zbl 0391.58014 · doi:10.1137/0135020 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.