## 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

### Keywords:

invariant measure; Lyapunov exponent

### Citations:

Zbl 0671.58019; Zbl 0497.58017
Full Text:

### References:

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