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

A measure-preserving transformation $T$ of a probability space is mixing if $\mu(A\cap T^{-n}B)\to\mu(A)\mu(B)$ as $n\to\infty$, and mixing of all orders if the analogous factorization holds for any finite number of sets at times whose successive separations tend to infinity. Since Rohlin's 1949 paper it has been open whether every mixing $\mathbb{Z}$-action (single invertible measure-preserving transformation) is mixing of all orders; this is one of the oldest unsolved problems in ergodic theory. The problem has an equivalent self-joinings formulation — a mixing transformation fails multiple mixing exactly when it admits a non-product, pairwise independent self-joining arising as a weak limit of off-diagonal graph self-joinings — and, following Ryzhikov, a minimal-self-joinings formulation asking whether the classes MSJ(2) and MSJ(3) coincide for $\mathbb{Z}$-actions, a counterexample to which would already refute Rohlin's problem.

Background

An automorphism $T$ of a Lebesgue probability space $(X,\mathcal{B},\mu)$ (a standard non-atomic probability space with countably generated $\sigma$-algebra, unique up to measure-preserving isomorphism) — that is, an invertible measurable map with $\mu(T^{-1}A)=\mu(A)$ — is called mixing (2-fold mixing) if for all measurable $A,B$, $\mu(A\cap T^{-n}B)\to\mu(A)\mu(B)\quad\text{as } n\to\infty.$ More generally, $T$ is $k$-fold mixing if for all measurable $A_0,\dots,A_{k-1}$ and all times with successive gaps $m_1,\dots,m_{k-1}\to\infty$, $\mu\bigl(A_0\cap T^{-m_1}A_1\cap\cdots\cap T^{-(m_1+\cdots+m_{k-1})}A_{k-1}\bigr)\to\prod_{i=0}^{k-1}\mu(A_i),$ and mixing of all orders if this holds for every $k$ (in Rohlin's original terminology, mixing of multiplicity $k-1$). Mixing of order $k+1$ implies mixing of order $k$, and whether the converse implications hold is the problem. V. A. Rokhlin introduced these notions in 1949 while searching for metric invariants that would distinguish ergodic automorphisms of compact commutative groups — ergodic meaning that every $T$-invariant set has measure $0$ or $1$ — with the same spectrum; he proved that all such algebraic systems are mixing of all orders, found that his new invariants therefore did not separate them, and posed the general question, which has remained open for over 75 years (Ryzhikov 2024).

For actions of $\mathbb{Z}^d$, $d\ge 2$, the answer is negative: Ledrappier (1978) constructed a mixing $\mathbb{Z}^2$-action — the two coordinate shifts on a closed, locally constrained subgroup of $(\mathbb{Z}/2\mathbb{Z})^{\mathbb{Z}^2}$ — that is not mixing of all orders, and variants of the construction separate each order of mixing from the next. This finite-range-constraint mechanism is provably unavailable in one dimension: de la Rue (2006) showed that no one-dimensional analogue of Ledrappier's construction can be mixing but not 3-fold mixing. Even the weakest open case — does mixing imply 3-fold mixing for a single transformation? — and the analogous problem for weakly mixing transformations (weakly mixing meaning $\frac{1}{n}\sum_{k<n}\lvert\mu(A\cap T^{-k}B)-\mu(A)\mu(B)\rvert\to 0$, a property intermediate between ergodicity and mixing) are open (Ryzhikov 2024). Bergelson and Zelada (2023) proved that every mixing action of a countable abelian group is 'almost' mixing of all orders in the sense of $\mathcal{R}$-limits, a Ramsey-theoretic mode of convergence; hence any counterexample must fail multiple mixing only along time-configurations invisible to that notion of largeness.

Many positive classes are known: mixing Gaussian systems, i.e. shift systems whose coordinates form a stationary Gaussian process (Leonov 1960); mixing transformations with singular spectral type, meaning that the maximal spectral measure of the associated unitary operator is singular (Host 1991); mixing rank-one transformations — systems approximated, up to sets of arbitrarily small measure, by a single Rokhlin tower (a tower is the orbit segment $x, Tx, \dots, T^{h-1}x$ of a base set) — where Kalikow (1984) proved that 2-fold mixing implies 3-fold mixing and Ryzhikov (1993) extended this to all orders and to finite-rank actions; and actions of positive local rank (Ryzhikov 2000), local rank measuring how well a system can be approximated by towers covering a definite fraction of the space. For flows, multiple mixing is known for horocycle flows — the flows on quotients of $\mathrm{SL}(2,\mathbb{R})$ generated by its unipotent one-parameter subgroup — (Marcus 1978), where Ratner's (1983) analysis of their self-joinings is the prototype of the joining approach, and for typical minimal components of broad classes of locally Hamiltonian surface flows (Fayad and Kanigowski 2016).

The self-joinings framework reformulates the problem. A joining of $n$ copies of $T$ is a measure on $X^n$, invariant under $T\times\cdots\times T$, whose one-coordinate marginals equal $\mu$; when all copies are the same system it is a self-joining. The basic examples are the product $\mu^{\otimes n}$ and the graph (off-diagonal) self-joinings $(x\mapsto (x,T^{a_1}x,\dots,T^{a_{n-1}}x))_*\mu$. If $T$ is mixing, then along any family of times with pairwise separations tending to infinity, subsequential weak limits of graph self-joinings are self-joinings whose every two-coordinate marginal is $\mu\otimes\mu$ (pairwise independence); $T$ is mixing of all orders exactly when all such limits are the product measure. Host's theorem and much of the modern theory work through this reformulation: excluding nontrivial pairwise independent self-joinings establishes multiple mixing. del Junco and Rudolph (1987) studied systems whose pairwise independent self-joinings are all product (the class PID), and their question of whether a mixing zero-entropy transformation can lack this property is still open; King (1992) proved that triviality of pairwise independent joinings at order 4 already forces it at every order. Following Ryzhikov's memoir (2021), write $T\in\mathrm{MSJ}(n)$ — minimal self-joinings of order $n$ — if every ergodic $n$-fold self-joining of $T$ other than $\mu^{\otimes n}$ has some two-coordinate face equal to a shifted diagonal $\Delta_{T^i}=(x\mapsto (x,T^i x))_*\mu$; transformations in $\mathrm{MSJ}(2)$ are weakly mixing with trivial centralizer (the only measure-preserving transformations commuting with $T$ are its own powers $T^k$, $k\in\mathbb{Z}$). For $\mathbb{Z}$-actions it is open whether the classes $\mathrm{MSJ}(2)$ and $\mathrm{MSJ}(3)$ coincide. Within the class $\mathrm{MSJ}(2)$ this question is equivalent to Rohlin's problem, and any $\mathbb{Z}$-action in $\mathrm{MSJ}(2)\setminus\mathrm{MSJ}(3)$ would automatically be mixing yet fail 3-fold mixing — by a theorem of Glasner, Host and Rudolph (1992) forcing such a transformation to be mixing, combined with Ryzhikov's theorem that $\mathrm{MSJ}(2)$ plus 3-fold mixing implies minimal self-joinings of all orders — and would thus solve Rohlin's problem negatively. For non-commutative group actions the classes genuinely differ: examples with $\mathrm{MSJ}(2)\neq\mathrm{MSJ}(3)\neq\mathrm{MSJ}(4)$ are presented in Ryzhikov's memoir (2021).

Problem Statement

Rohlin's multiple mixing problem (open since 1949). Let $T$ be a mixing (2-fold mixing) invertible measure-preserving transformation of a Lebesgue probability space $(X,\mathcal{B},\mu)$. Must $T$ be mixing of all orders — that is, for every $k\ge 3$, all measurable $A_0,\dots,A_{k-1}$, and all integer times with successive gaps $m_1,\dots,m_{k-1}\to\infty$, does $\mu\bigl(A_0\cap T^{-m_1}A_1\cap\cdots\cap T^{-(m_1+\cdots+m_{k-1})}A_{k-1}\bigr)\to\prod_{i=0}^{k-1}\mu(A_i)?$ Equivalently, in the self-joinings formulation (a self-joining of $T$ is a measure on $X^k$, invariant under $T\times\cdots\times T$, whose every one-coordinate marginal is $\mu$; it is pairwise independent if every two-coordinate marginal is $\mu\otimes\mu$): must every pairwise independent self-joining of a mixing transformation that arises as a weak-* limit of graph self-joinings $(x\mapsto (x,T^{a_1}x,\dots,T^{a_{k-1}}x))_*\mu$ with pairwise separated, escaping times be the product measure $\mu^{\otimes k}$? In Ryzhikov's minimal-self-joinings formulation — where $T\in\mathrm{MSJ}(n)$ means that every ergodic $n$-fold self-joining of $T$ other than $\mu^{\otimes n}$ has some two-coordinate face equal to a shifted diagonal $(x\mapsto (x,T^i x))_*\mu$: do the classes $\mathrm{MSJ}(2)$ and $\mathrm{MSJ}(3)$ coincide for $\mathbb{Z}$-actions — and, equivalently within the class $\mathrm{MSJ}(2)$, is every mixing transformation with minimal self-joinings of order 2 mixing of all orders?

References

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