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 ag (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.