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

A problem posed by V. A. Rohlin in 1949 asks whether, for a probability-preserving transformation $T$ (a single measure-preserving $\mathbb{Z}$-action), two-fold mixing already forces mixing of every order: must the multiple correlation $\mu(A_0\cap T^{n_1}A_1\cap\cdots\cap T^{n_1+\cdots+n_k}A_k)$ converge to the product $\prod_j\mu(A_j)$ whenever the successive time gaps diverge, for every $k$? The problem is open already for three sets. For $\mathbb{Z}^d$-actions with $d\ge 2$ the answer is negative (Ledrappier's 1978 three-dot Markov field), and many special classes of mixing systems are known to be mixing of all orders (singular spectrum, finite rank, Gaussian and Poisson systems, systems with ergodic homoclinic group, unipotent and related parabolic flows), but the general single-transformation case has resisted more than 75 years of work. A dedicated November 2024 survey by Ryzhikov and the surrounding 2024–2026 literature confirm that the problem remains open.

Background

Ergodic theory studies measure-preserving transformations: invertible measurable maps $T$ of a probability space $(X,\mathcal{B},\mu)$ with $\mu(T^{-1}A)=\mu(A)$ for all measurable $A$. Iterating $T$ turns $(X,\mu)$ into a stationary random process, and classical mixing properties quantify how independence emerges between the present and the remote future. The weakest standard notion, ergodicity, says time averages converge to space averages. Strong mixing (two-fold mixing) says $\mu(A\cap T^{n}B)\to\mu(A)\mu(B)$ for all measurable $A,B$ as $n\to\infty$: the event 'now' becomes asymptotically independent of the event at a distant time.

Rohlin refined this into a hierarchy. A transformation $T$ has $k$-fold mixing ($k\ge 1$; also called mixing of order $k$, a property involving $k+1$ sets; in the sets-counting terminology the same property is called $(k+1)$-fold mixing, so strong mixing is 'two-fold mixing' in that usage) if for all measurable $A_0,A_1,\dots,A_k$,

$$\mu\big(A_0\cap T^{n_1}A_1\cap T^{n_1+n_2}A_2\cap\cdots\cap T^{n_1+\cdots+n_k}A_k\big)\ \to\ \mu(A_0)\,\mu(A_1)\cdots\mu(A_k)$$

whenever $n_1\to+\infty$ and each successive gap $n_{j+1}\to+\infty$; equivalently, along any sequence of time tuples whose pairwise differences all diverge. The case $k=1$ is strong mixing itself, and $T$ is mixing of all orders if it has $k$-fold mixing for every $k$. The question is whether this hierarchy collapses at the bottom for a single transformation.

The problem originates in Rohlin's 1949 paper on endomorphisms of compact commutative groups (Izv. Akad. Nauk SSSR Ser. Mat. 13 (1949), 329–340; English translation in Amer. Math. Soc. Transl. Series 2, 64 (1967), 244–252). Rohlin introduced the $k$-fold mixing invariants while searching for metric invariants that could distinguish dynamical systems with the same spectrum — a spectral problem of the time that was later resolved by other means through entropy theory. He proved that ergodic automorphisms of compact commutative groups (which preserve Haar measure) do mix of all orders, and posed the general question of whether the invariants coincide, i.e. whether every mixing transformation mixes of every order. The question has remained open for more than 75 years and is among the most famous unsolved problems in ergodic theory; it is open already at the first nontrivial level $k=2$ (three sets), i.e. whether two-fold mixing implies three-fold mixing in the terminology that counts sets.

The answer is known to be negative for actions of higher-rank lattices. Ledrappier (1978) constructed a $\mathbb{Z}^2$-action — the 'three-dot' system, a stationary mod-2 Markov field on $\mathbb{Z}^2$ defined by the closed shift-invariant subgroup of $\{0,1\}^{\mathbb{Z}^2}$ in which $x_{(0,0)}+x_{(1,0)}+x_{(0,1)}=0$ mod 2 — which is mixing (and zero entropy) but fails three-fold mixing: the parity relation forces $x_{(0,0)}+x_{(n,0)}+x_{(0,n)}=0$ for every $n$, so a triple correlation stays bounded away from the product of the measures even though all pairwise time differences diverge. As Ryzhikov's survey recounts, this compact-group-counterexample strategy succeeds for $\mathbb{Z}^n$-actions for every $n\ne 1$; the rank-one lattice $\mathbb{Z}$ is precisely the case where it breaks down.

Many positive classes are known, each importing a structural hypothesis. Host (1991) proved that a mixing transformation with singular spectrum — one whose spectral measure of maximal type for the Koopman operator $f\mapsto f\circ T$ is singular with respect to Lebesgue measure — is mixing of all orders, via his theorem that pairwise independent joinings of such systems are independent (a joining is an invariant measure on a product space with the right marginals; joinings are the standard tool for studying multiple correlations). Kalikow (1984) proved that two-fold mixing implies three-fold mixing for rank-one transformations (systems approximable by a single cutting-and-stacking tower), extended to finite-rank actions by Ryzhikov (1993); in the wider class of positive local rank (approximation by towers covering a fixed positive fraction of the space) only conditional results are known (Ryzhikov 2000). Leonov (1960) proved all-orders mixing for mixing Gaussian systems (systems built functorially from a stationary Gaussian process, where Wick-cumulant computations reduce multiple correlations to products of covariances); mixing Poisson suspensions behave similarly; and Ryzhikov (2019), building on Gordin's observation that an ergodic homoclinic group forces mixing, proved that an automorphism whose homoclinic group $H(T)=\{S : T^{-n}ST^{n}\to\mathrm{Id}\}$ — the group of automorphisms conjugate to the identity at infinity — contains an ergodic element is mixing of all orders, a theorem covering the Gaussian and Poisson cases. In homogeneous dynamics, Marcus (1978) proved the horocycle flow (the flow along horocycle directions on a quotient of the hyperbolic plane) is mixing of all degrees, Mozes (1992) extended all-orders mixing to Lie group actions, Ratner's joining rigidity for horocycle flows (1983) yields the property for flows with the Ratner property (a quantitative orbit-shearing property of flows abstracted from her rigidity theorems), Fayad and Kanigowski (2016) proved it for a class of conservative surface flows, and Kanigowski and Ravotti (2024) added the class of locally uniformly shearing flows (a rigorous formulation of 'mixing via shearing', covering smooth surface flows and smooth time-changes of unipotent flows, i.e. flows on homogeneous spaces generated by unipotent one-parameter subgroups such as the horocycle flow). Ryzhikov's surveys also note the striking heuristic that 'the slower the mixing, the more reliably multiple mixing holds': failure of multiple mixing would require very fast mixing.

Several modern results delimit the question without closing it. Bergelson and Zelada (2024) proved that every strongly mixing action of a countable abelian group is 'almost strongly mixing of all orders': higher-order correlations do factor along a Ramsey-theoretic notion of large time sets (R-limits), so any failure could hide only along exceptionally sparse time configurations. De la Rue (2004) showed that the relative version of the question fails — there exists an extension that is relatively two-fold mixing but not relatively three-fold mixing — while the absolute problem stands. Ryzhikov's 2024 problem list records that a hypothetical mixing transformation failing three-fold mixing would have Lebesgue spectrum (a Koopman operator whose maximal spectral type is Lebesgue measure) and infinite rank (no approximation by finitely many towers), and would simultaneously bear on other famous open questions about joinings and spectrum. As of the 2024 dedicated surveys and the 2024–2026 literature, the general $\mathbb{Z}$-action problem remains open.

Problem Statement

Let $T$ be a measure-preserving transformation (an invertible measure-preserving map, i.e. a probability-preserving $\mathbb{Z}$-action) of a probability space $(X,\mathcal{B},\mu)$, with no restriction on the space or the transformation. Say that $T$ has $k$-fold mixing ($k\ge 1$) if for all measurable sets $A_0,A_1,\dots,A_k$,

$$\mu\big(A_0\cap T^{n_1}A_1\cap T^{n_1+n_2}A_2\cap\cdots\cap T^{n_1+\cdots+n_k}A_k\big)\ \to\ \mu(A_0)\,\mu(A_1)\cdots\mu(A_k)$$

whenever $n_1\to+\infty$ and each successive gap $n_{j+1}\to+\infty$ (equivalently, along every sequence of time tuples whose pairwise differences all diverge); $k=1$ is ordinary strong mixing, and $T$ is mixing of all orders if it has $k$-fold mixing for every $k\ge 1$.

Rohlin's problem (1949): does every mixing measure-preserving transformation have $k$-fold mixing for every $k\ge 2$ — i.e. does two-fold mixing imply mixing of all orders? The problem is open already at $k=2$, that is, whether every mixing $T$ satisfies

$$\mu\big(A_0\cap T^{n_1}A_1\cap T^{n_1+n_2}A_2\big)\ \to\ \mu(A_0)\,\mu(A_1)\,\mu(A_2)$$

for all measurable $A_0,A_1,A_2$ as $n_1,n_2\to+\infty$. A complete answer is either a proof that mixing implies $k$-fold mixing for all $k$ (for all probability-preserving transformations), or an explicit mixing transformation together with sets and admissible time sequences along which some higher-order correlation fails to converge to the product of the measures. No restriction on the class of transformations, the probability space, or the order $k$ is permitted; the known negative results for $\mathbb{Z}^d$-actions with $d\ge 2$ do not bear on this single-transformation question.

References

1. V. A. Rokhlin, On endomorphisms of compact commutative groups, Izv. Akad. Nauk SSSR Ser. Mat. 13:4 (1949), 329–340; English translation: Metric properties of endomorphisms of compact commutative groups, Amer. Math. Soc. Translations Series 2, 64 (1967), 244–252. DOI 10.1090/trans2/064/08, https://doi.org/10.1090/trans2/064/08; zbMATH 0183.03602, https://zbmath.org/?q=an%3A0183.03602
2. F. Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant, C. R. Acad. Sci. Paris Sér. A 287 (1978), 561–563. zbMATH 0387.60084, https://zbmath.org/?q=an%3A0387.60084
3. V. P. Leonov, The use of the characteristic functional and semi-invariants in the ergodic theory of stationary processes, Sov. Math. Dokl. 1 (1960), 878–881; Russian original: Dokl. Akad. Nauk SSSR 133:3 (1960), 523–526. zbMATH 0141.15401, https://zbmath.org/?q=an%3A0141.15401
4. B. Host, Mixing of all orders and pairwise independent joinings of systems with singular spectrum, Israel Journal of Mathematics 76 (1991), 289–298. DOI 10.1007/BF02773866, https://doi.org/10.1007/BF02773866
5. S. A. Kalikow, Twofold mixing implies threefold mixing for rank one transformations, Ergodic Theory and Dynamical Systems 4 (1984), 237–259. DOI 10.1017/S014338570000242X, https://doi.org/10.1017/S014338570000242X
6. V. V. Ryzhikov, Joinings and multiple mixing of finite rank actions, Functional Analysis and Its Applications 27:2 (1993), 128–140. DOI 10.1007/BF01085983, https://doi.org/10.1007/BF01085983; zbMATH 0853.28011, https://zbmath.org/?q=an%3A0853.28011
7. V. V. Ryzhikov, Rokhlin's multiple mixing problem in the class of positive local rank actions, Functional Analysis and Its Applications 34:1 (2000), 73–75. DOI 10.1007/BF02467072, https://doi.org/10.1007/BF02467072; zbMATH 0957.28009, https://zbmath.org/?q=an%3A0957.28009
8. V. V. Ryzhikov, Weakly homoclinic groups of ergodic actions, Transactions of the Moscow Mathematical Society 80 (2019), 83–94. DOI 10.1090/mosc/289, https://doi.org/10.1090/mosc/289; zbMATH 1440.37004, https://zbmath.org/?q=an%3A1440.37004
9. B. Marcus, The horocycle flow is mixing of all degrees, Inventiones Mathematicae 46 (1978), 201–209. DOI 10.1007/BF01390274, https://doi.org/10.1007/BF01390274
10. S. Mozes, Mixing of all orders of Lie groups actions, Inventiones Mathematicae 107 (1992), 235–241. DOI 10.1007/BF01231889, https://doi.org/10.1007/BF01231889
11. M. Ratner, Horocycle flows, joinings and rigidity of products, Annals of Mathematics (2) 118 (1983), 277–313. DOI 10.2307/2007030, https://doi.org/10.2307/2007030
12. B. Fayad, A. Kanigowski, Multiple mixing for a class of conservative surface flows, Inventiones Mathematicae 203 (2016), 555–614. DOI 10.1007/s00222-015-0596-6, https://doi.org/10.1007/s00222-015-0596-6
13. V. V. Ryzhikov, Multiple mixing, 75 years of Rokhlin's problem, arXiv:2411.07234 (2024). https://arxiv.org/abs/2411.07234
14. V. V. Ryzhikov, Multiple mixing in ergodic theory (in Russian), arXiv:2404.06476 (2024). https://arxiv.org/abs/2404.06476 (this is the earlier Russian version of the survey)
15. V. V. Ryzhikov, Unsolved problems on joinings, multiple mixing, spectrum, and rank, arXiv:2405.03641 (2024). https://arxiv.org/abs/2405.03641
16. V. Bergelson, R. Zelada, Strongly mixing systems are almost strongly mixing of all orders, Ergodic Theory and Dynamical Systems 44:6 (2024), 1489–1530. DOI 10.1017/etds.2023.63, https://doi.org/10.1017/etds.2023.63
17. A. Kanigowski, D. Ravotti, Multiple mixing for parabolic systems, arXiv:2410.13686 (2024). https://arxiv.org/abs/2410.13686
18. T. de la Rue, An extension which is relatively twofold mixing but not threefold mixing, Colloquium Mathematicum 101 (2004), 271–277. DOI 10.4064/cm101-2-9, https://doi.org/10.4064/cm101-2-9
19. Z. J. Wang, Multiple mixing and multiple fractional cohomological equation: semisimple setting, arXiv:2605.21173 (2026). https://arxiv.org/abs/2605.21173