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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
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