On the Nonuniqueness of the Invariant Probability for I.I.D. Random Logisitc Maps

Abstract

Let $\{X_n\}^{\infty}_0$ be a Markov chain with values in $[0,1]$ generated by the iteration of random logistic maps defined by $X_{n+1}=f_{C_{n+1}}(X_n)\equiv C_{n+1}X_n(1-X_n)$, $n=0,1,2,\ldots\,$, with $\{C_n\}^{\infty}_1$ being independent and identically distributed random variables with values in $[0,4]$ and independent of $X_0$. This paper provides a class of examples where $C_i$ take only two values $\lambda$ and $\mu$ such that there exist two distinct invariant probability distributions $\pi_0$ and $\pi_1$ supported by the open interval $(0,1)$. This settles a question raised by R. N. Bhattacharya.