Ulam meets Turing: constructing quadratic maps with non-computable SRB measures

TL;DR

This paper demonstrates that for certain logistic maps, the long-term statistical behavior can be non-computable, revealing fundamental limits of numerical methods like Monte Carlo in dynamical systems.

Contribution

It constructs explicit examples of simple non-linear maps with non-computable SRB measures, bridging Ulam's and Turing's ideas in dynamical systems.

Findings

Existence of parameters with non-computable statistical distributions

Numerical methods can fail for simple logistic maps

Almost all orbits share the same non-computable distribution

Abstract

In 1946, S. Ulam invented Monte Carlo method, which has since become the standard numerical technique for making statistical predictions for long-term behaviour of dynamical systems. We show that this, or in fact any other numerical approach can fail for the simplest non-linear discrete dynamical systems given by the logistic maps $f_{a}(x)=ax(1-x)$ of the unit interval. We show that there exist computable real parameters $a\in (0,4)$ for which almost every orbit of $f_a$ has the same asymptotical statistical distribution in $[0,1]$, but this limiting distribution is not Turing computable.