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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(f_{a}(x)=ax(1-x)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mi>a</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of the unit interval. We show that there exist computable real parameters <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(a\in (0,4)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for which almost every orbit of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(f_a\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> has the same asymptotical statistical distribution in [0,&#xa0;1], but this limiting distribution is not Turing computable.</p>

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Ulam Meets Turing: Constructing Quadratic Maps with Non-Computable Physical Measures

  • 期刊论文

  • 发表日期：

    2026年2月2日

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  • Cristóbal Rojas,
  • Michael Yampolsky

摘要

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 \(f_{a}(x)=ax(1-x)\) f a ( x ) = a x ( 1 - x ) of the unit interval. We show that there exist computable real parameters \(a\in (0,4)\) a ∈ ( 0 , 4 ) for which almost every orbit of \(f_a\) f a has the same asymptotical statistical distribution in [0, 1], but this limiting distribution is not Turing computable.