Rohlin's problem: does strong mixing imply mixing of all orders for measure-preserving $\mathbb{Z}$-actions?

Rohlin's problem (1949) asks whether a strongly mixing (2-mixing) measure-preserving transformation of a probability space is automatically mixing of every finite order, i.e., whether all multiple correlations along time parameters with pairwise diverging gaps factorize as products of the measures. The question is the oldest open problem in the hierarchy of mixing properties for single transformations: Ledrappier (1978) showed it has a negative answer for $\mathbb{Z}^2$-actions, positive answers are known only under structural restrictions (compact-group algebraic systems, singular spectrum, finite or positive local rank), and Bergelson and Zelada (2020/2023) proved that every strongly mixing action of a countable abelian group is 'almost' strongly mixing of all orders in a sharp combinatorial sense while explicitly leaving the classical $\mathbb{Z}$-case open. As of August 2026 the problem remains open.

Background

A measure-preserving system is a probability space $(X,\mathcal{A},\mu)$ together with a measure-preserving transformation $T:X\to X$ (a $\mathbb{Z}$-action, iterated as $T^n$). The system is strongly mixing (or 2-mixing) if for all measurable $A_0,A_1\in\mathcal{A}$ one has $\lim_{n\to\infty}\mu(A_0\cap T^{-n}A_1)=\mu(A_0)\mu(A_1)$, meaning that events decorrelate as their time separation grows. For a single transformation, mixing is the strongest decorrelation property expressible by a pair of times, but Szemerédi-type multiple recurrence naturally involves correlations of three or more events. The system is $(\ell+1)$-mixing (mixing of order $\ell+1$) if for all $A_0,\dots,A_\ell\in\mathcal{A}$ and all sequences of integer $\ell$-tuples whose pairwise differences tend to infinity in absolute value, $\mu(A_0\cap T^{-n_k^{(1)}}A_1\cap\cdots\cap T^{-n_k^{(\ell)}}A_\ell)\to\prod_{j=0}^{\ell}\mu(A_j)$; mixing of all orders means $(\ell+1)$-mixing for every $\ell\ge 2$. Rohlin asked in 1949 whether strong mixing implies mixing of all orders for $\mathbb{Z}$-actions, proving the implication for endomorphisms of compact commutative groups. The question is delicate: Ledrappier constructed a strongly mixing $\mathbb{Z}^2$-action (commuting automorphisms $T,S$ of a compact abelian group endowed with its Haar measure, i.e., the translation-invariant probability measure) that is not 3-mixing, with $\mu(A\cap T^{2^n}A\cap S^{2^n}A)\not\to\mu(A)^3$; higher-rank lattice actions can support such algebraic obstructions, while no analogue is known for a single transformation. For $\mathbb{Z}$-actions the implication is known only under structural hypotheses: mixing transformations with singular spectrum — meaning that the maximal spectral measure of the unitary operator $f\mapsto f\circ T$ on $L^2(\mu)$ is singular with respect to Lebesgue measure (Host 1991, via the theory of joinings, i.e., invariant measures on product spaces that couple two systems); finite-rank actions — rank being the complexity notion measuring how many towers $A, TA, \dots, T^{n-1}A$ (stacks of successive images of one set covering almost all of the space) suffice to approximate the system, rank-one being the simplest case (Kalikow 1984 proved twofold mixing implies threefold mixing for rank-one transformations; Ryzhikov 1993 extended this to all orders for finite-rank actions); and positive-local-rank actions, a one-sided local analogue of the rank condition (Ryzhikov 2000). Bergelson and Zelada (arXiv 2020, journal 2023) proved a universal relaxation: for any countable abelian group $G$, every strongly mixing $G$-action is 'almost' strongly mixing of all orders, in that the sets of $\ell$-tuples where the $(\ell+1)$-fold correlation is within $\epsilon$ of the product enjoy a combinatorial largeness property ($\tilde{\Sigma}^*_\ell$, via $\mathcal{R}$-limits based on Ramsey's theorem) forcing factorization along sequences that avoid a provably small exceptional set. The authors note these conclusions are less sharp than the tailored Ledrappier-type analyses of Arenas-Carmona, Berend and Bergelson (2008, 2019), and they explicitly restate that Rohlin's problem for $\mathbb{Z}$-actions remains unsolved.

Problem Statement

Rohlin's problem for $\mathbb{Z}$-actions. Let $(X,\mathcal{A},\mu)$ be a probability space and $T:X\to X$ a measure-preserving transformation such that $\mu(A_0\cap T^{-n}A_1)\to\mu(A_0)\mu(A_1)$ for all $A_0,A_1\in\mathcal{A}$ (strong mixing / 2-mixing). Is $T$ necessarily mixing of all orders? That is, for every $\ell\ge 2$, all $A_0,\dots,A_\ell\in\mathcal{A}$, and all sequences $(n_k^{(1)})_{k\in\mathbb{N}},\dots,(n_k^{(\ell)})_{k\in\mathbb{N}}$ of integers with $n_k^{(i)}\to\infty$ for each $i$ and $|n_k^{(i)}-n_k^{(j)}|\to\infty$ for all $i\ne j$, does

$$\mu\left(A_0\cap T^{-n_k^{(1)}}A_1\cap\cdots\cap T^{-n_k^{(\ell)}}A_\ell\right)\longrightarrow\prod_{j=0}^{\ell}\mu(A_j)?$$

The problem is to settle this question in full generality: either prove that every strongly mixing measure-preserving $\mathbb{Z}$-action is mixing of all orders, or construct a strongly mixing $\mathbb{Z}$-action that fails to be 3-mixing (exhibiting measurable sets and a sequence of pairs $(n_k,m_k)$ with $n_k\to\infty$, $m_k\to\infty$, $m_k-n_k\to\infty$ along which the triple correlation fails to converge to the product of the measures). No structural restriction may be assumed: no hypothesis on entropy, spectrum, rank, or algebraic structure may be imposed, and the statement must hold for measure-preserving transformations of standard Lebesgue probability spaces (invertibility is not at issue, since every measure-preserving transformation on such a space has a standard invertible extension).

References

1. Vitaly Bergelson and Rigoberto Zelada, Strongly mixing systems are almost strongly mixing of all orders, Ergodic Theory and Dynamical Systems (2023), DOI 10.1017/etds.2023.63, https://doi.org/10.1017/etds.2023.63 (preprint arXiv:2010.06146, https://arxiv.org/abs/2010.06146, v2 July 2021, 39 pp.)
2. V. A. Rohlin, On endomorphisms of compact commutative groups, Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya 13 (1949), 329-340 (Russian; no DOI exists for this 1949 article), zbMATH Zbl 0032.28402, https://zbmath.org/?q=an%3A3050054; English translation: V. A. Rokhlin, Metric properties of endomorphisms of compact commutative groups, American Mathematical Society Translations, Series II 64 (1967), 244-252, zbMATH Zbl 0183.03602, https://zbmath.org/?q=an%3A3291211
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6. V. V. Ryzhikov, Joinings and multiple mixing of finite rank actions, Functional Analysis and its Applications 27 (1993), 128-140, DOI 10.1007/BF01085983, https://doi.org/10.1007/BF01085983
7. V. V. Ryzhikov, Rokhlin's multiple mixing problem in the class of positive local rank actions, Functional Analysis and its Applications 34 (2000), 73-75, DOI 10.1007/BF02467072, https://doi.org/10.1007/BF02467072
8. L. Arenas-Carmona, D. Berend, and V. Bergelson, Ledrappier's system is almost mixing of all orders, Ergodic Theory and Dynamical Systems 28 (2008), 339-365, DOI 10.1017/S0143385707000727, https://doi.org/10.1017/S0143385707000727
9. L. Arenas-Carmona, D. Berend, and V. Bergelson, An almost mixing of all orders property of algebraic dynamical systems, Ergodic Theory and Dynamical Systems 39 (2019), 1211-1233, DOI 10.1017/etds.2017.60, https://doi.org/10.1017/etds.2017.60
10. Sohail Farhangi, Joel Moreira, and Rigoberto Zelada, Multipliers and Disjointness from Mixing, preprint submitted April 2026, arXiv:2604.12915, https://arxiv.org/abs/2604.12915
11. V. V. Ryzhikov, Multiple Mixing, Local Rank, and Joinings of Automorphisms of a Probability Space, Theory of Probability and its Applications 70 (2026), 614-622, DOI 10.1137/S0040585X97T992653, https://doi.org/10.1137/S0040585X97T992653