  We consider the simplest non-linear discrete dynamical systems, given by the
logistic maps $f_{a}(x)=ax(1-x)$ of the interval $[0,1]$. We show that there
exist real parameters $a\in (0,4)$ for which almost every orbit of $f_a$ has
the same statistical distribution in $[0,1]$, but this limiting distribution is
not Turing computable. In particular, the Monte Carlo method cannot be applied
to study these dynamical systems.
