Given the logistic map $x_{n+1} = r x_n(1-x_n)$, it is well-known that if $0 \le r\le 1$, then $\lim_{n\rightarrow\infty} x_n = 0$ regardless of the value of $x_0\in(0,1)$. What is the asymptotic rate of convergence to zero in this case, and why? In other words, how fast does it go to zero? Is $x_n = O(x_0^n)$?
This is a simpler version of what I actually want, which is the asymptotic rate of convergence, for constant $z\in(0,1/2]$, of the logistic map $x_{n+1} = z^2 x_n(\frac{2}{z}-x_n)$.