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

TL;DR

This paper demonstrates that certain simple logistic maps have statistical behaviors that are non Turing computable, making traditional Monte Carlo methods ineffective for analyzing these dynamical systems.

Contribution

It introduces a class of logistic maps with non-computable invariant distributions, revealing fundamental limitations of computational methods in dynamical systems analysis.

Findings

Existence of parameters with non-computable invariant measures

Almost all orbits share the same non-computable distribution

Monte Carlo methods cannot approximate these distributions

Abstract

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 $a\in (0,4)$ 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.