Random logistic maps. I (Q1582137)

The paper studies iterations of i.i.d. random logistic maps, i.e. it studies the sequence \(X_{n+1}=C_{n+1}X_n(1-X_n)\) where \(C_n, n=1,2,\dots\), is a sequence of i.i.d. random variables with values in \([0,4].\) The authors show that \(X_n\) tends to zero a.s. if \(E\ln C_1<0\) and in probability if \(E\ln C_1=0\). If \(E\ln C_1>0\) and \(E|\ln(4-C_1)|<\infty\), then there exists an invariant measure of the corresponding Markov chain whose properties are discussed.