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

Abstract

We consider the simplest non-linear discrete dynamical systems, given by the logistic maps fa(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 fa 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.2010 Mathematics Subject Classification. 68Q17 and 37E05.