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