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

Cristóbal Rojas, Michael Yampolsky

TL;DR

The paper addresses the reliability of Monte Carlo statistics for long-term dynamics by showing there exist computable parameters for the logistic map $f_a(x)=a x(1-x)$ with a unique physical (SRB) measure whose basin covers the whole domain, yet the limiting measure is not Turing computable. It develops a constructive Johnson–Hofbauer–Keller framework to encode noncomputable Halting information into the distribution on a countable family of periodic orbits, yielding a computable $a$ with a noncomputable SRB measure $\mu_a$. The main contributions are a concrete, constructive proof of noncomputability of the physical measure in this simple dynamical system and a detailed methodology for embedding noncomputable data into invariant measures via parabolic parameters and Cantor-like repellers. Practically, this reveals fundamental limits of Monte Carlo predictions for chaotic systems and motivates further study of computability and complexity in dynamical statistics, including the prevalence of such phenomena in higher-dimensional settings.

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.

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

TL;DR

The paper addresses the reliability of Monte Carlo statistics for long-term dynamics by showing there exist computable parameters for the logistic map with a unique physical (SRB) measure whose basin covers the whole domain, yet the limiting measure is not Turing computable. It develops a constructive Johnson–Hofbauer–Keller framework to encode noncomputable Halting information into the distribution on a countable family of periodic orbits, yielding a computable with a noncomputable SRB measure . The main contributions are a concrete, constructive proof of noncomputability of the physical measure in this simple dynamical system and a detailed methodology for embedding noncomputable data into invariant measures via parabolic parameters and Cantor-like repellers. Practically, this reveals fundamental limits of Monte Carlo predictions for chaotic systems and motivates further study of computability and complexity in dynamical statistics, including the prevalence of such phenomena in higher-dimensional settings.

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 of the unit interval. We show that there exist computable real parameters for which almost every orbit of has the same asymptotical statistical distribution in , but this limiting distribution is not Turing computable.
Paper Structure (8 sections, 4 theorems, 36 equations, 1 figure)

This paper contains 8 sections, 4 theorems, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the Main Theorem
  4. Setting the Stage
  5. Constructing Non Computable Physical Measures
  6. Johnson's construction
  7. Proof of the main result
  8. Conclusion

Key Result

Proposition 3.1

Suppose, for $a\in[0,4]$ the map $f_a$ has a non-repelling periodic orbit of period $p$: Let Then $\mu$ is the unique physical measure of $f_a$ (so, in particular, the non-repelling orbit is unique); and uniformly on a set of full Lebesgue measure in $[0,1]$.

Figures (1)

  • Figure 1: Some iterates of $f\equiv f_a$ for $a\in(c,4]$ restricted to the appropriate intervals (we drop the subscript $a$ for simplicity in all notations in the figure).

Theorems & Definitions (8)

  • Definition 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Definition 3.1
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof