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A zero-entropy system without a smooth positive-volume model
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Abstract
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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Notes
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Zero entropy does not guarantee a smooth positive-volume model (Lean scope note) (lean/docs/152.md, release of 6 October 2026). The release's scope note says the formalized statement covers a finite-entropy system with no smooth positive-volume model, and that the paper's stronger zero-entropy conclusion is outside it. [family-level]
- OpenAI, SmoothObstruction.lean (comparator statement) (openai/math, October 2026). This is the comparator statement
SmoothObstruction.main: there is a standard nonatomic ergodic system withFiniteKSEntropyand¬ HasSmoothPositiveVolumeModel, stated with asorrybody as a challenge. - OpenAI, OAI/Dynamics/SmoothObstruction/Main.lean (openai/math, October 2026). This module states and proves the finite-entropy theorem
mainusing imported modules (Counting,BoundaryDensity,Obstruction), and zero entropy is not part of the formal statement.
Possibly relevant literature (found by the agent, not checked)
- B. Fayad, M. Saprykina, A. Windsor, Nonstandard smooth realizations of Liouville rotations (arXiv:math/0608322, August 2006). Its introduction states the general question "whether every finite entropy automorphism of a Lebesgue space has a smooth realization", which is the question Theorem 1.1 answers negatively for smooth positive densities.
- B. Fayad, A. Katok, Constructions in elliptic dynamics (arXiv:math/0501362, January 2005). This is a survey of the Anosov–Katok approximation-by-conjugation method for building smooth volume-preserving systems with prescribed ergodic properties, which is background for the paper's zero-entropy obstruction.
Digests and expositions
- Latent Seminar, Zero entropy and the obstruction to smooth positive-volume models (2026-10-06). This exposition explains the smooth-density requirement, the switching estimate and symbolic compatibility obstruction, and the finite-entropy formalization scope, explicitly disclaiming independent verification of the complete proofs. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 152
Independent sites; their content is not reviewed here.
Cite
@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}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/152.md. The Lean statements have not been checked by people against the paper's claims.
Family 152: Zero entropy does not guarantee a smooth positive-volume model
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.
This is the only paper in this family.