## On iterations of $$1-\alpha x^2$$ on $$(-1,1)$$.(English)Zbl 0597.58016

In the paper the mapping $$F(x;a)=1-ax^ 2$$, $$-1<x<1$$, where $$a$$ is a parameter in the interval $$(0,2)$$ is considered. The paper has three parts. The first one is introductory. Existence of attractive cycles is studied in the second part and it is proved that there exists a set $$S\subset (0,2)$$ of positive Lebesgue measure such that for $$a\in S$$ the mapping $$x\to F(x;a)$$ has no attractive cycles. In the third part the authors prove that for almost all $$a\in S$$, $$x\to F(x;a)$$ has an absolutely continuous invariant measure. Moreover, they prove that the density function of that measure belongs to $$L^ p$$ for $$p<2$$. The proof also shows that for $$2-a_ 0$$ small enough the set of $$a\in (a_ 0,2)$$ for which $$F(x;a)$$ has an attractive cycle is open and dense in $$S$$.
Reviewer: J.Šiška

### MSC:

 37A10 Dynamical systems involving one-parameter continuous families of measure-preserving transformations 26A18 Iteration of real functions in one variable 28A75 Length, area, volume, other geometric measure theory 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior

### Keywords:

invariant measure; iterations; attractive cycles
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