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Real bounds, ergodicity and negative Schwarzian for multimodal maps. (English) Zbl 1073.37043
J. Am. Math. Soc. 17, No. 4, 749-782 (2004); erratum ibid. 20, No. 1, 267-268 (2007).
For unimodal maps on the interval many interesting results have been proved using methods which work only in the unimodal case. Here, the authors try to provide methods in order to obtain similar results also for multimodal maps.
The maps considered in this paper are multimodal maps having finitely many critical points and satisfying that all critical points are nonflat. Note that the map may have inflection points. Using the methods developed in this paper, several interesting results (known in the unimodal case) are proved. Among these results are the nonexistence of wandering intervals, distortion theorems, negative Schwarzian derivative for certain return maps, and certain ergodic properties.
Although the presentation is sometimes very technical the paper is well written.
Reviewer: Peter Raith (Wien)

MSC:
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37A25 Ergodicity, mixing, rates of mixing
37E10 Dynamical systems involving maps of the circle
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