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Regularity and other properties of absolutely continuous invariant measures for the quadratic family. (English) Zbl 0770.58021

In [Ergodic Theory Dyn. Syst. 8, No. 1, 93-109 (1988; Zbl 0671.58019)] the first author gave a new proof of Jakobson’s theorem [M. V. Jakobson, Commun. Math. Phys. 81, 39-88 (1981; Zbl 0497.58017)]. In the current paper the dynamical system \(f_ \alpha\) defined on the unit interval, \(f_ \alpha(x)=1-\alpha x^ 2\), is considered for the parameters \(\alpha\) for which there is an absolutely continuous invariant measure (a.c.i.m) \(\nu_ \alpha\). It is shown that the density of the invariant measure and Lyapunov exponent associated to the dynamical system \((f_ \alpha,\nu_ \alpha)\) are continuous at the so called Misiurewicz points. It is also proved that the Lyapunov exponent is positive and the dynamical system \((f_ \alpha,\nu_ \alpha)\) is exact.
The specified results are obtained via the study of the continuity of the invariant measures and the spectrum of the Perron-Frobenius operator for maps of class BV. (A mapping \(T\) is of class BV if it maps \(I\setminus S\) to \(I\), \(S\) being a closed set of measure 0).

MSC:

37A99 Ergodic theory
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[1] Benedicks, M., Carleson, L.: On iterations of 1-ax 2 on (, 1). Ann. Math. (2)122 (1), 1–25 (1985) · Zbl 0597.58016
[2] Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics, vol.470, Berlin, Heidelberg, New York: Springer-Verlag 1975 · Zbl 0308.28010
[3] Hofbauer, F., Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z.180, 119–140 (1982) · Zbl 0485.28016
[4] Krzy\.zewski, K., Szlenk, W.: On invariant measures for expanding differentiable mappings. Studia Math.23, 83–92 (1969) · Zbl 0176.00901
[5] Krzy\.zewski, K.: On connection between expanding mappings and Markov chains. Bull. Acad. Pol. Sci. Ser. Math., vol.19, 291–293 (1971) · Zbl 0208.52003
[6] Keller, G.: On the Rate of Convergence to Equilibrium in One-Dimensional Systems. Commun. Math. Phys.96, 181–193 (1984) · Zbl 0576.58016
[7] Ledrappier, F.: Some properties of absolutely continuous measures on an interval. Ergodic Theory Dyn. Syst.1, 77–93 (1981) · Zbl 0487.28015
[8] Jakobson, M.V.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys.81, 39–88 (1981) · Zbl 0497.58017
[9] Rychlik, M.: Bounded variation and invariant measures. Studia Math. t.76, 69–80 (1983) · Zbl 0575.28011
[10] Rychlik, M.: Regularity of the Metric Entropy for Expanding Maps. Preprint, IAS, 1988
[11] Rychlik, M.: Another proof of Jakobson’s theorem and related results. Engrdic Theory Dyn. Syst.8, 93–109 (1988) · Zbl 0671.58019
[12] Young, L.-S.: Decay of Correlations for Certain Quadratic Maps. Preprint, UCLA, 1991
[13] Ziemian, K.: Almost sure invariance principle for some maps of an interval. Engrdic Theory Dyn. Syst.5, 625–640 (1985) · Zbl 0604.60031
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