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Hyperbolicity and invariant measures for general $$C^ 2$$ interval maps satisfying the Misiurewicz condition. (English) Zbl 0702.58020
Summary: We show that piecwise $$C^ 2$$ mappings f on [0,1] or $$S^ 1$$ satisfying the so-called Misiurewicz conditions are globally expanding (in the sense defined below) and have absolute continuous invariant probability measures of positive entropy. We do not need assumptions on the Schwarzian derivative of these maps. Instead we need the conditions that f is piecewise $$C^ 2$$, that all critical points of f are “non-flat”, and that f has no periodic attractors. Our proof gives an algorithm to verify this last condition. Our result implies the result of M. Misiurewicz in Publ. Math., Inst. Hautes Étud. Sci. 53, 17-51 (1981; Zbl 0477.58020) (where only maps with negative Schwarzian derivatives are considered). Moreover, as a byproduct, the present paper implies (and simplifies the proof of) the results of R. Mañe in ibid. 100, 495-524 (1985; Zbl 0583.58016), Erratum ibid. 112, 721-724 (1987; Zbl 0628.58032), who considers general $$C^ 2$$ maps (without conditions on the Schwarzian derivative), and restricts attention to points whose forward orbit stays away from the critical points. One of the main complications will be that in this paper we want to prove the existence of invariant measures and therefore have to consider points whose iterations come arbitrarily close to critical points. Misiurewicz deals with this problem using an assumption on the Schwarzian derivative of the map. This assumption implies very good control of the non-linearity of $$f^ n$$, even for high n. In order to deal with this non-linearity, without an assumption on the Schwarzian derivative, we use the tools of W. de Melo and the author [Ann. Math., II. Ser. 129, No.3, 519-546 (1989) and Bull. Am. Math. Soc., New. Ser. 18, No.2, 159-162 (1988; Zbl 0651.58019)]. It will turn out that the estimates we obtain are so precise that the existence of invariant measures can be proved in a very simple way (in some sense much simpler than used by M. Misiurewicz in the paper cited above). The existence of these invariant measures under such general conditions was already conjectured a decade ago.

##### MSC:
 37E99 Low-dimensional dynamical systems 37D99 Dynamical systems with hyperbolic behavior 37A99 Ergodic theory 28D05 Measure-preserving transformations
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##### References:
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