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Math-2694 · mathematics · Dynamical systems

Zero-entropy probability system with no smooth positive-volume model

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Does there exist an invertible ergodic measure-preserving transformation on a standard nonatomic probability space, with zero Kolmogorov-Sinai entropy, which is not measure-theoretically isomorphic to any C-infinity diffeomorphism preserving a strictly positive smooth probability density on any finite-dimensional compact smooth manifold, including nonorientable manifolds and manifolds with boundary?

Short English statement

Does there exist an invertible ergodic measure-preserving transformation on a standard nonatomic probability space, with zero Kolmogorov-Sinai entropy, which is not measure-theoretically isomorphic to any C-infinity diffeomorphism preserving a strictly positive smooth probability density on any finite-dimensional compact smooth manifold, including nonorientable manifolds and manifolds with boundary?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2694-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2694

Manuscript 1 (family 152, 2026-09-25): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Zero-entropy probability system with no smooth positive-volume model

OpenAI's claim

A zero-entropy invertible ergodic nonatomic probability system is claimed to have no realization by any finite-dimensional smooth diffeomorphism preserving a strictly positive smooth density. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
OpenAI claims an ergodic invertible probability-preserving transformation $(X,\mu,T)$ on a standard nonatomic probability space with \[ h_\mu(T)=0, \] which is not conjugate measurably after deleting invariant sets of measure zero to any smooth diffeomorphism of a compact manifold of finite dimension whose preserved probability measure has smooth everywhere-positive density. The exclusion covers nonorientable models as well as manifolds with a smooth boundary. The measure class is positive smooth volume; arbitrary singular invariant measures are not covered.

Outline of the argument

Manuscript 1 (family 152, 2026-09-25): For smooth positive volume, a dyadic switching estimate joins the future of one nearby point to the past of another with controlled output measure and summable cell complexity. Derivative-log bounds and finite boundary-value calculations establish that estimate; the constructed system violates the resulting measure-theoretic switching constraint in every finite dimension.

Formal verification

For family 152, the scope page documents the following selections. The linked obstruction only requires finite measure entropy. The manuscript's sharper zero-entropy property is beyond that declaration, whereas exclusion of smooth positive-volume models includes compact manifolds with boundary. This entry does not assert that the selected declarations formalize every part of manuscript 1's main conclusion. Comparator files: SmoothObstruction.lean. Declarations: OAI.SmoothObstruction.main.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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