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

How to lose at Monte Carlo: a simple dynamical system whose typical statistical behavior is non computable