Does there exist a non-Bernoulli K-automorphism all of whose nontrivial splittings consist of one Bernoulli and one non-Bernoulli factor?

The weak Pinsker theorem (Austin, 2018) states that every ergodic measure-preserving automorphism of positive entropy admits, for every $\varepsilon>0$, a splitting into a Bernoulli shift and a residual factor of entropy less than $\varepsilon$. The same paper asks, as Question 16.8, an extremal structural question that lies strictly beyond that theorem: does there exist a non-Bernoulli K-automorphism — an ergodic system with completely positive entropy that is not isomorphic to a Bernoulli shift — such that every nontrivial splitting of it consists of exactly one Bernoulli and one non-Bernoulli factor? Such a system would show that Bernoulli factors can be woven into every product decomposition of a K-system yet never appear in pairs. The sibling question on finitary weak Pinsker splittings was answered negatively by Gabor (2019), but the present question remains open, with no construction or impossibility result known.

Background

An automorphism $(X,\mu,T)$ is an invertible, measure-preserving transformation of a standard probability space; it is ergodic if every $T$-invariant measurable set has measure 0 or 1. The Kolmogorov–Sinai entropy $h(T)$ is the standard numerical invariant measuring the rate at which the system produces randomness. A Bernoulli shift is the shift map on a two-sided sequence space $A^{\mathbb{Z}}$ over a finite or countable alphabet $A$, equipped with a product (independent-coordinate) measure; its entropy may be finite or infinite, and Ornstein's isomorphism theorem classifies Bernoulli shifts up to isomorphism by entropy alone. A K-automorphism (Kolmogorov system, equivalently a system with completely positive entropy) is an ergodic automorphism for which every nontrivial invariant factor has positive entropy. Every Bernoulli shift is K, but the converse fails: classical examples of Ornstein, and the $T,T^{-1}$ 'random walk in random scenery' skew product — the transformation that reads out the scenery of a static random environment along the trajectory of a simple random walk on $\mathbb{Z}$ (proved K by Meilijson and non-Bernoulli by Kalikow; see Thouvenot's 2023 survey) — are K but not Bernoulli.

A splitting of $(X,\mu,T)$ is an isomorphism of it with a direct product $(Y_1,\nu_1,S_1)\times(Y_2,\nu_2,S_2)$; equivalently, a pair of independent $T$-invariant sub-$\sigma$-algebras of $X$ that together generate the full $\sigma$-algebra of $X$ (modulo null sets). A splitting is nontrivial when neither factor is the one-point system, and it has a Bernoulli factor when one of the two factors is isomorphic to a Bernoulli shift.

Historically, Pinsker conjectured that every ergodic automorphism of positive entropy splits as a zero-entropy system times a K-system; Ornstein refuted this. Thouvenot then introduced the weak Pinsker property — for every $\varepsilon>0$ the system splits as a Bernoulli shift times a residual factor of entropy less than $\varepsilon$ — and asked whether every ergodic automorphism has it. The source paper proves that every ergodic automorphism of positive entropy does; in particular, every such system has a nontrivial splitting with a Bernoulli factor.

Since Bernoulli factors are thus unavoidable in positive entropy, the basic open issue becomes the structure of the lattice of splitting factors. Kalikow constructed a non-Bernoulli transformation with two splitting $\sigma$-algebras having trivial intersection, so splitting factors need not nest; Thouvenot's survey records that equal-entropy splitting factors of a fixed system must nevertheless be isomorphic, and that many structure questions remain. An old heuristic of Ornstein asserts that the classification of factors inside a fixed Bernoulli shift should mirror the complexity of classifying general K-automorphisms (Ornstein 1975; the examples of Hoffman 1999). Question 16.8 of the source paper sits at the extreme end of this circle of ideas: it asks whether there is a non-Bernoulli K-automorphism in which Bernoulli factors are unavoidable but never duplicated — every nontrivial splitting consists of exactly one Bernoulli and one non-Bernoulli factor. Austin remarks that the question 'seems to lie strictly beyond the weak Pinsker property': the theorem produces many Bernoulli splittings but gives no control over the full lattice of splittings.

Problem Statement

Does there exist a non-Bernoulli K-automorphism $(X,\mu,T)$ with the property that any non-trivial splitting of it consists of one Bernoulli and one non-Bernoulli factor? (Austin, 'Measure concentration and the weak Pinsker property', Question 16.8.) Here a splitting is an isomorphism of $(X,\mu,T)$ with a direct product $(Y_1,\nu_1,S_1)\times(Y_2,\nu_2,S_2)$ of measure-preserving systems — equivalently, a pair of $T$-invariant, independent sub-$\sigma$-algebras that together generate the full $\sigma$-algebra of $X$ (modulo null sets); the splitting is non-trivial when neither factor is the one-point system; and 'one Bernoulli and one non-Bernoulli factor' means that exactly one of the two factors is isomorphic to a Bernoulli shift while the other is not. Since a product of two Bernoulli systems is Bernoulli, no such system exists exactly when every non-Bernoulli K-automorphism admits a nontrivial splitting into two non-Bernoulli factors.

References

1. Tim Austin, Measure concentration and the weak Pinsker property, Publications Mathématiques de l'IHÉS 128 (2018), 1–119. DOI: 10.1007/s10240-018-0098-3; arXiv:1705.00302. https://doi.org/10.1007/s10240-018-0098-3
2. Jean-Paul Thouvenot, The Brin Prize works of Tim Austin, Journal of Modern Dynamics 19 (2023), 835–845. DOI: 10.3934/jmd.2023024. https://doi.org/10.3934/jmd.2023024
3. Uri Gabor, On the failure of Ornstein theory in the finitary category, Transactions of the American Mathematical Society (published online 2024). DOI: 10.1090/tran/8776; arXiv:1909.11453. https://doi.org/10.1090/tran/8776
4. Steven Kalikow, Non-intersecting splitting σ-algebras in a non-Bernoulli transformation, Ergodic Theory and Dynamical Systems 32 (2012), 691–705. DOI: 10.1017/S0143385711000034. https://doi.org/10.1017/S0143385711000034
5. Donald S. Ornstein, Factors of Bernoulli shifts, Israel Journal of Mathematics 21 (1975), 145–153. DOI: 10.1007/BF02760792. https://doi.org/10.1007/BF02760792
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7. Tim Austin, Eli Glasner, Jean-Paul Thouvenot, Benjamin Weiss, An ergodic system is dominant exactly when it has positive entropy, Ergodic Theory and Dynamical Systems 43 (2023), 3216–3230 (online 2022). DOI: 10.1017/etds.2022.69; arXiv:2112.03800. https://doi.org/10.1017/etds.2022.69
8. Séverin Benzoni, Introduction to weak Pinsker filtrations, arXiv:2504.00681 (2025). https://arxiv.org/abs/2504.00681
9. Tim Austin, Lewis Bowen, Christopher Shriver, Algebraic dynamical systems from LDPC codes satisfy a strong negation of the weak Pinsker property, Journal of the Institute of Mathematics of Jussieu 24 (2025), 2317–2385. DOI: 10.1017/S147474802510100X; arXiv:2312.17387. https://doi.org/10.1017/S147474802510100X
10. Benjamin Weiss, Some Generic Properties of Processes, arXiv:2507.21620 (2025). https://arxiv.org/abs/2507.21620
11. Tim Austin, Some Recent Developments in Structural Ergodic Theory, Proceedings of the International Congress of Mathematicians 2026, Volume 5: Invited Lectures (Sections 9–11), SIAM (2026), 3–23. DOI: 10.1137/25m1805564. https://doi.org/10.1137/25m1805564