Rokhlin's multiple mixing problem: does mixing imply mixing of all orders for measure-preserving automorphisms?

Rokhlin's multiple mixing problem, open since 1949, asks whether every mixing (2-fold mixing) measure-preserving automorphism of a probability space is automatically mixing of every finite order $k\ge3$: if correlations of two events decay to independence at large time lags, must correlations of events at $k$ widely separated times also factorize? The formulation follows V. V. Ryzhikov's 2024 survey of the problem. All known positive results require structure beyond mixing (rank one or finite rank, local rank above 1/2, shearing mechanisms for parabolic flows, algebraic and spectral-gap hypotheses), while counterexamples exist only for $\mathbb{Z}^d$-actions with $d\ge2$ (Ledrappier's 3-dot system), for mixing relative to a factor, and in approximate 'almost mixing of all orders' senses. No proof and no counterexample is known for single automorphisms: the problem, already open in its first nontrivial case $k=3$, remains open.

Background

Ergodic theory studies measure-preserving dynamical systems. An automorphism is an invertible transformation $T$ of a probability space $(X,\mathcal{B},\mu)$ such that $\mu(T^{-1}A)=\mu(A)$ for every measurable set $A$. The system is called mixing (equivalently, 2-fold mixing) if events become asymptotically independent as the time lag between them grows: $\mu(A\cap T^{n}B)\to\mu(A)\,\mu(B)$ as $n\to\infty$ for all measurable $A$ and $B$. Mixing is a property of pairs of times; V. A. Rokhlin introduced in 1949 its higher-order analogue, the hierarchy of $k$-fold mixing. An automorphism is $k$-fold mixing ($k\ge2$) if events observed at $k$ times whose successive lags all tend to infinity become jointly independent: $\mu\big(A_0\cap T^{m_1}A_1\cap T^{m_1+m_2}A_2\cap\cdots\cap T^{m_1+\cdots+m_{k-1}}A_{k-1}\big)\ \longrightarrow\ \prod_{j=0}^{k-1}\mu(A_j)$ whenever $m_1,\dots,m_{k-1}\to\infty$ jointly. Rokhlin was studying automorphisms of compact commutative groups (invertible endomorphisms of compact abelian groups preserving Haar probability measure, a classical source of measure-preserving systems); his motivation was spectral — he wanted invariants able to distinguish metrically non-isomorphic systems whose induced unitary operators on $L^2(X,\mu)$ have the same spectrum, in particular the same Lebesgue (purely absolutely continuous) spectrum. He proved that ergodic automorphisms of compact commutative groups are mixing of all orders, and asked whether the hierarchy collapses for an arbitrary automorphism: does ordinary mixing already imply $k$-fold mixing for every $k$? The question has been open for more than 75 years. The problem is known to be false for actions of larger groups: F. Ledrappier's '3-dot system' of 1978, the shift action on the closed subgroup of $\{0,1\}^{\mathbb{Z}^2}$ defined by the constraint $x_{s}+x_{s+e_1}+x_{s+e_2}=0 \bmod 2$ (where $e_1,e_2$ are the standard generators of $\mathbb{Z}^2$), is a $\mathbb{Z}^2$-action by commuting automorphisms that is 2-fold mixing but not 3-fold mixing, and similar counterexamples exist for all $\mathbb{Z}^d$-actions with $d\ge2$. For a single automorphism, however, T. de la Rue proved that Ledrappier-type constructions cannot be adapted to one-dimensional (i.e. $\mathbb{Z}$-indexed) stationary processes, so any counterexample for automorphisms needs a genuinely new mechanism. In the positive direction, all results cover classes with additional structure: S. A. Kalikow proved that twofold mixing implies threefold mixing for rank-one transformations (automorphisms generated by a cutting-and-stacking construction on a single tower, that is, a stack $B, TB, \dots, T^{h-1}B$ of successive images of one base set $B$; these are the simplest objects of ergodic theory), a result later extended to finite-rank transformations; it is known that no counterexample can have local rank exceeding $1/2$ (local rank $\beta$ means the automorphism can be approximated, on a set of measure arbitrarily close to $\beta$, by a single tower); and A. Kanigowski and D. Ravotti proved that parabolic flows — continuous-time systems in which nearby orbits separate at a subexponential, typically polynomial, rate — satisfying a locally uniform shearing mechanism (orbit segments starting at different heights drift apart at locally uniform speeds) are mixing of all orders; their class includes smooth mixing flows on surfaces and smooth time-changes of unipotent flows (flows generated by unipotent linear maps, that is, linear maps whose only eigenvalue is $1$, acting on compact homogeneous spaces). Approximate versions hold universally: V. Bergelson and R. Zelada showed that strongly mixing actions of countable abelian groups are almost strongly mixing of all orders (in an $\mathcal{R}$-limit sense, a Ramsey-theoretic notion of convergence along combinatorially rich subsequences), and L. Arenas-Carmona, D. Berend and V. Bergelson showed that even Ledrappier's system is almost mixing of all orders once a small set of exceptional times is excluded; on the other hand the relative analogue fails, since T. de la Rue constructed an extension that is relatively twofold mixing but not relatively threefold mixing. The unifying technical language for these questions is the theory of joinings: a self-joining of $T$ is a measure on a finite product $X\times\cdots\times X$, invariant under $T\times\cdots\times T$, whose projection onto each coordinate is $\mu$; a non-product self-joining with product-like marginals is exactly how a failure of higher-order mixing would be witnessed. Rokhlin's problem is the central open question of this circle of ideas; surrounding it stand related open problems collected in Ryzhikov's recent surveys — the del Junco–Rudolph problem on zero-entropy mixing automorphisms without property $S_p$ (property $S_p$ rules out nontrivial self-joinings with trivial $(p-1)$-dimensional marginals), the question of whether the minimal-self-joinings classes $MSJ(2)$ and $MSJ(3)$ can be separated ($MSJ(k)$ denotes the class, introduced by D. Rudolph, of automorphisms whose $k$-fold self-joinings are all of the trivial expected form), and Banach's problem on simple Lebesgue spectrum (from the problem list in Halmos's book: does there exist a measure-preserving automorphism with simple Lebesgue spectrum?) — and a hypothetical automorphism in $MSJ(2)\setminus MSJ(3)$ would have Lebesgue spectrum and, if its spectrum is simple, would resolve the Banach, Rokhlin and del Junco–Rudolph problems simultaneously.

Problem Statement

Rokhlin's multiple mixing problem (1949). Let $T$ be a measure-preserving automorphism of a probability space $(X,\mathcal{B},\mu)$, and suppose that $T$ is mixing, that is, $\mu(A\cap T^{n}B)\ \longrightarrow\ \mu(A)\,\mu(B)\qquad(n\to\infty)$ for all measurable sets $A,B\in\mathcal{B}$. Must $T$ then be $k$-fold mixing for every $k\ge3$? That is: for every $k\ge3$, every choice of measurable sets $A_0,A_1,\dots,A_{k-1}$, and every sequence of integer lags with $m_1,\dots,m_{k-1}\to\infty$ jointly, must $\mu\big(A_0\cap T^{m_1}A_1\cap T^{m_1+m_2}A_2\cap\cdots\cap T^{m_1+\cdots+m_{k-1}}A_{k-1}\big)\ \longrightarrow\ \mu(A_0)\,\mu(A_1)\cdots\mu(A_{k-1})\ ?$ The first nontrivial case $k=3$ is already open: for general automorphisms no proof is known that mixing implies 3-fold mixing, and no mixing automorphism failing 3-fold or higher-order mixing has ever been constructed. In the language of joinings, a negative answer would in particular yield a mixing automorphism admitting a nontrivial 3-fold self-joining (a $T\times T\times T$-invariant measure on $X^3$ whose coordinate projections all equal $\mu$) whose two-coordinate marginals are the product measure $\mu\times\mu$. The problem concerns a single automorphism (a $\mathbb{Z}$-action); for $\mathbb{Z}^d$-actions with $d\ge2$ the analogous statement is false, as shown by Ledrappier's 3-dot system.

References

1. V. A. Rokhlin, On endomorphisms of compact commutative groups, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 13 (1949), 329-340, Zbl 0032.28402, https://zbmath.org/0032.28402 (Russian original; English translation: V. A. Rohlin, Metric properties of endomorphisms of compact commutative groups, American Mathematical Society Translations Series 2, vol. 64 (1967), pp. 244-252, DOI 10.1090/trans2/064/08, https://doi.org/10.1090/trans2/064/08)
2. F. Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant, C. R. Acad. Sci. Paris Sér. A 287 (1978), 561-563, Zbl 0387.60084, https://zbmath.org/0387.60084
3. 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
4. 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
5. T. de la Rue, 2-fold and 3-fold mixing: why 3-dot-type counterexamples are impossible in one dimension, arXiv:math/0603154 (2006), https://arxiv.org/abs/math/0603154
6. L. Arenas-Carmona, D. Berend, 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
7. V. Bergelson, R. Zelada, Strongly mixing systems are almost strongly mixing of all orders, Ergodic Theory and Dynamical Systems 44 (2024), 1489-1530, DOI 10.1017/etds.2023.63, https://doi.org/10.1017/etds.2023.63 (preprint arXiv:2010.06146)
8. V. V. Ryzhikov, Multiple mixing in ergodic theory, arXiv:2404.06476 (2024), DOI 10.48550/arXiv.2404.06476, https://arxiv.org/abs/2404.06476 (survey source of this problem record)
9. V. V. Ryzhikov, Unsolved problems on joinings, multiple mixing, spectrum, and rank, arXiv:2405.03641 (2024), https://arxiv.org/abs/2405.03641
10. V. V. Ryzhikov, Multiple mixing, 75 years of Rokhlin's problem, arXiv:2411.07234 (2024), https://arxiv.org/abs/2411.07234 (English companion of the Russian survey cited above)
11. A. Kanigowski, D. Ravotti, Multiple mixing for parabolic systems, arXiv:2410.13686 (2024), https://arxiv.org/abs/2410.13686
12. Z. J. Wang, Multiple mixing and multiple fractional cohomological equation: semisimple setting, arXiv:2605.21173 (2026), https://arxiv.org/abs/2605.21173
13. V. V. Ryzhikov, Around Kalikow's theorem on multiple mixing, arXiv:2607.11852 (2026), https://arxiv.org/abs/2607.11852