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).


37A99 Ergodic theory
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