Keller, Gerhard Exponents, attractors and Hopf decompositions for interval maps. (English) Zbl 0715.58020 Ergodic Theory Dyn. Syst. 10, No. 4, 717-744 (1990). Summary: Our main results, specialized to unimodal interval maps T with negative Schwarzian derivative, are the following: (1) There is a set \(C_ T\) such that the \(\omega\)-limit of Lebesgue-a.e. point equals \(C_ T\). \(C_ T\) is a finite union of closed intervals or it coincides with the closure of the critical orbit. (2) There is a constant \(\lambda_ T\) such that \(\lambda_ T=\limsup_{n\to \infty}(1/n)\log | (T^ n)'(x)|\) for Lebesgue- a.e. x. (3) \(\lambda_ T>0\) if and only if T has an absolutely continuous invariant measure of positive entropy. (4) \(\lambda_ T\geq \inf \{p^{-1}\log | (T^ p)'(z)|:\) \(T^ pz=z\}\), i.e. uniform hyperbolicity on periodic points implies \(\lambda_ T>0\), and \(\lambda_ T<0\) implies the existence of a stable periodic orbit. Cited in 3 ReviewsCited in 22 Documents MSC: 37A99 Ergodic theory 28D20 Entropy and other invariants Keywords:Hopf-decomposition; regular Markov systems; unimodal interval maps; Schwarzian derivative PDF BibTeX XML Cite \textit{G. Keller}, Ergodic Theory Dyn. Syst. 10, No. 4, 717--744 (1990; Zbl 0715.58020) Full Text: DOI References: [1] DOI: 10.1007/BF01212280 · Zbl 0595.58028 · doi:10.1007/BF01212280 [2] Misiurewicz, Publ. Math. I.H.E.S. 53 pp 17– (1981) · Zbl 0477.58020 · doi:10.1007/BF02698686 [3] DOI: 10.1007/BF00534111 · doi:10.1007/BF00534111 [4] DOI: 10.1007/BF02760884 · Zbl 0422.28015 · doi:10.1007/BF02760884 [5] DOI: 10.2307/1971501 · Zbl 0708.58007 · doi:10.2307/1971501 [6] DOI: 10.1007/BF01982351 · Zbl 0429.58012 · doi:10.1007/BF01982351 [7] Collet, Iterated Maps on the Interval as Dynamical Systems (1983) [8] Brudno, Trans. Mosc. Math. Soc. 2 pp 127– (1983) [9] Brudno, Ergodic Theory and Related Topics pp 23– (1981) [10] Breiman, Probability (1968) [11] Blokh, Ergod. Th. & Dynam. Sys. 9 pp none– (1989) [12] Aaronson, J. d’Analyse Math. 39 pp 203– (1981) [13] Krenge, Ergodic Theorems. (1985) · doi:10.1515/9783110844641 [14] DOI: 10.1007/BF01308670 · Zbl 0712.28008 · doi:10.1007/BF01308670 [15] Keller, Invariant measures and Lyapunov exponents (1987) [16] DOI: 10.2307/2001395 · Zbl 0686.58027 · doi:10.2307/2001395 [17] DOI: 10.1007/BF02096761 · Zbl 0702.58034 · doi:10.1007/BF02096761 [18] DOI: 10.1007/BF00334191 · Zbl 0578.60069 · doi:10.1007/BF00334191 [19] Hofbauer, Ergod. Th. & Dynam. Sys. 1 pp 159– (1981) [20] DOI: 10.1007/BF02761854 · Zbl 0456.28006 · doi:10.1007/BF02761854 [21] DOI: 10.1007/BF01303262 · Zbl 0433.54009 · doi:10.1007/BF01303262 [22] van Strien, Hyperbolicity and invariant measures for general C (1988) [23] Shaw, Z. Naturf. A 36 pp 80– (1981) · doi:10.1515/zna-1981-0115 [24] Nowicki, Hyperbolicity properties of C (1988) [25] Nowicki, Fundamenta Math. 126 pp 27– (1985) [26] de Melo, A structure theorem in one dimensional dynamics (1986) 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.