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∈ (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.

While most polynomial Julia sets are computable, it has been recently shown [12] that there exist non-computable Julia sets. The proof was non-constructive, and indeed there were doubts as to whether specific examples of parameters with non-computable Julia sets could be constructed. It was also unknown whether the non-computability proof can be extended to the filled Julia sets. In this paper we give an answer to both of these questions, which were the main open problems concerning the computability of polynomial Julia sets.