1 · The essentials
A system can have zero entropy—a measure of information generated by its evolution—and still resist a particular kind of smooth description. The manuscript constructs one such system. Its evolution is reversible, and it cannot be split into two unchanging parts that both have positive probability. Yet no change of description that preserves its measurable dynamics turns it into the smooth model considered here. Such a model would live on a compact space of finite dimension, evolve smoothly, and preserve a smoothly distributed probability that is strictly positive everywhere. The same example rules out every finite dimension, not just a chosen one. The main result has a Lean formalization, although the supplied material does not specify its detailed scope.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine trying to redraw a motion on a smooth, bounded stage while keeping its probability rules intact. The result says that one zero-entropy motion cannot be redrawn in the required way on any of the permitted finite-dimensional stages, however many dimensions are allowed.
Why it matters
The result separates zero entropy from the existence of a smooth positive-volume model: the former does not guarantee the latter. Its force comes from using a single example to exclude all finite dimensions. The conclusion concerns a specific class of smooth, probability-preserving models, not every possible smooth description.
2 · Going further
The precise question is whether zero Kolmogorov–Sinai entropy guarantees a smooth positive-volume model for an ergodic invertible transformation of a standard nonatomic probability space. Here, a model means a measurable conjugacy to a diffeomorphism of a compact finite-dimensional manifold that preserves a strictly positive smooth probability density. Measurable conjugacy requires the two evolutions to agree under a probability-preserving measurable change of description, up to sets of measure zero.
The manuscript constructs a transformation satisfying the stated probability-space, invertibility, ergodicity, and entropy conditions but having no such model. Its quantifiers are important: a single transformation excludes every finite dimension. The abstract explicitly includes nonorientable manifolds and manifolds with smooth boundary among the excluded model spaces.
The supplied abstract describes the construction's outcome but does not give the mechanism of the construction or the argument excluding smooth models. It therefore does not support a more detailed account of the argument. Only the main manuscript is supplied, so no relationship to companion manuscripts can be described. The main result has a Lean formalization, but the supplied material does not identify which subsidiary claims or extensions its scope covers.
Useful notions
- Probability spaces and measure-preserving evolution
- Invertibility and ergodicity
- Kolmogorov–Sinai entropy
- Measurable conjugacy
- Smooth manifolds and diffeomorphisms
What was known before / what is new
Before
The source frames the issue through the relationship between zero entropy and smooth positive-volume models. It gives no account of earlier results, historical milestones, or a previously posed open problem.
New
The manuscript presents a zero-entropy, ergodic, invertible transformation with no smooth positive-volume model of the specified kind. One example excludes all finite dimensions, including models on nonorientable manifolds and manifolds with smooth boundary.
Limits and assumptions
The constructed system is on a standard nonatomic probability space and is ergodic, invertible, and of zero Kolmogorov–Sinai entropy. The exclusion concerns measurable conjugacy to diffeomorphisms on compact finite-dimensional manifolds preserving strictly positive smooth probability densities. The supplied material does not establish corresponding exclusions under weaker smoothness assumptions, different density conditions, or outside this class of spaces. The main result has a Lean formalization; its detailed coverage is not specified here, so formal verification of every stated extension should not be inferred. The abstract supplies neither the construction details nor a list of remaining open questions.
Mini glossary
- Ergodicity —
- A measure-preserving system is ergodic when every measurable set unchanged by the evolution has probability zero or one.
- Kolmogorov–Sinai entropy —
- A measure of the rate at which a probability-preserving system generates information through its evolution.
- Measurable conjugacy —
- An invertible, probability-preserving measurable correspondence that matches two systems' evolutions, disregarding sets of measure zero.
- diffeomorphism —
- An invertible map between smooth manifolds for which both the map and its inverse have derivatives of every order.
- Smooth positive-volume model —
- In this result, a measurably conjugate system given by a diffeomorphism on a compact finite-dimensional manifold preserving a strictly positive smooth probability density.
Keywords : Zero entropy · Measurable conjugacy · Smooth positive-volume models
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
A zero-entropy system without a smooth positive-volume model
25/09/2026We 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.