∑math20

Dynamical systems and ergodic theory · Result 152

Zero entropy does not guarantee a smooth positive-volume model

1 paperLean

Constructs a zero-entropy ergodic invertible transformation of a standard nonatomic probability space that is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. One example rules out every finite dimension.

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The problem

What is being asked, where it comes from, and why it is hard. No proof needed.

▶Introduction to the problemin preparation
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The solution proposed by OpenAI

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Papers 1

First page of A zero-entropy system without a smooth positive-volume model

A zero-entropy system without a smooth positive-volume model

OpenAI · September 25, 2026

We construct an ergodic invertible transformation of a standard nonatomic probability space with zero Kolmogorov–Sinai entropy that has no smooth positive-volume model. More precisely, it is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. A single example excludes every finite dimension, including models on nonorientable manifolds and manifolds with smooth boundary.

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@misc{OAI:A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026,
  author = {{OpenAI}},
  title = {{A zero-entropy system without a smooth positive-volume model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/paper.pdf}{OAI:A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The paper asks whether a measure-preserving system can be represented by a smooth diffeomorphism preserving positive smooth volume. The formalized supporting result constructs an ergodic invertible transformation of a standard nonatomic probability space with finite Kolmogorov–Sinai entropy that has no such model on any compact finite-dimensional manifold, including manifolds with smooth boundary. The comparison is measurable conjugacy after discarding null sets.

The linked statement gives finite entropy; the paper's stronger zero-entropy conclusion is outside this statement.

Comparator links

ResultComparator statement
Finite-entropy system without a smooth positive-volume modelSmoothObstruction.lean

Scope note on GitHub →