Rohlin's 1949 multifold mixing problem asks whether 2-fold mixing implies 3-fold mixing for a measure-preserving transformation; it is one of the oldest open problems in ergodic theory. Ledrappier (1978) showed the implication fails for Z2\mathbb{Z}^2-actions through his zero-entropy "3-dot" random field, in which the shifted copies are pairwise independent but not jointly independent. De la Rue (2006) proved that this 3-dot mechanism cannot be transplanted to one-dimensional stationary processes: a 3-dot-type self-joining (third coordinate letterwise a function f(ξi,ξi′)f(\xi_i,\xi'_i) of the other two) forces periodicity or entropy at least log⁡2\log 2. The paper leaves open the central question whether a zero-entropy, 2-fold mixing stationary process on a finite alphabet can admit a pairwise-independent 3-fold self-joining that differs from the product measure at all — in particular one whose third coordinate is an arbitrary measurable function φ(ξ,ξ′)\varphi(\xi,\xi') of the other two. The restricted case where ξ0′′\xi''_0 depends on only finitely many coordinates was subsequently resolved negatively (such joinings force positive entropy or periodicity), but the general measurable-φ\varphi case and the fully unconstrained case remain open, as confirmed by expert surveys in 2021 and 2024. A negative answer would resolve Rohlin's 3-fold mixing problem; a positive answer would produce the first known non-trivial pairwise-independent joining in the zero-entropy weakly mixing class.

Background

A stationary process is a random sequence ξ=(ξi)i∈Z\xi=(\xi_i)_{i\in\mathbb{Z}} taking values in a finite alphabet A\mathbb{A} whose distribution μ\mu is invariant under the shift on AZ\mathbb{A}^{\mathbb{Z}}; equivalently, a measure-preserving system together with a generating partition. The process is 2-fold mixing (mixing of order 2, ordinary mixing) if for all measurable sets A,BA,B, μ(A∩σ−nB)→μ(A)μ(B)\mu(A\cap\sigma^{-n}B)\to\mu(A)\mu(B) as n→∞n\to\infty, and 3-fold mixing if the triple correlations satisfy μ(A∩σ−nB∩σ−mC)→μ(A)μ(B)μ(C)\mu(A\cap\sigma^{-n}B\cap\sigma^{-m}C)\to\mu(A)\mu(B)\mu(C) whenever nn and m−nm-n both tend to infinity. Kolmogorov–Sinai entropy measures the asymptotic information production per time step; zero entropy means the process is deterministic in the sense that its entire future is measurable with respect to its past. Rohlin asked in 1949 whether 2-fold mixing implies 3-fold mixing for a single transformation; despite decades of work the question remains open (Ryzhikov 2021, 2024).

A powerful reformulation uses joinings. A joining of two stationary processes is a stationary probability measure on the product space whose two marginals are the given distributions; a 3-fold self-joining of ξ\xi is a stationary measure λ\lambda on AZ×AZ×AZ\mathbb{A}^{\mathbb{Z}}\times\mathbb{A}^{\mathbb{Z}}\times\mathbb{A}^{\mathbb{Z}} whose three marginals all equal μ\mu. It is pairwise independent if each two-dimensional marginal is the product μ⊗μ\mu\otimes\mu, and trivial (product) if λ=μ⊗μ⊗μ\lambda=\mu\otimes\mu\otimes\mu. If a 2-fold mixing process failed 3-fold mixing, the off-diagonal joinings of (ξ,σnξ,σ2nξ)(\xi,\sigma^n\xi,\sigma^{2n}\xi) would accumulate, along a subsequence, on a pairwise-independent self-joining that is not the product; moreover, de la Rue (2006) shows the failure can be reduced (via the Pinsker σ\sigma-algebra, the maximal zero-entropy factor) to the zero-entropy case. Hence: if no zero-entropy 2-fold mixing stationary process admits a pairwise-independent non-product 3-fold self-joining, then 2-fold mixing implies 3-fold mixing.

Ledrappier's 3-dot system shows this implication genuinely fails in dimension two: a Z2\mathbb{Z}^2-indexed random field with ξi,j=ξi−1,j+ξi,j−1 mod 2\xi_{i,j}=\xi_{i-1,j}+\xi_{i,j-1}\bmod 2 that is 2-fold mixing, has zero entropy, and is not 3-fold mixing; its three shifted copies form a pairwise-independent, non-product joining in which each coordinate is the pointwise sum of the other two. Classical results prove that no analogous phenomenon exists in large one-dimensional classes: Kalikow (1984) showed mixing rank-one transformations (built by the cutting-and-stacking construction with one tower) are 3-fold mixing, Ryzhikov (1993) extended this to finite-rank mixing actions of all mixing orders, and Host (1991) proved that systems with singular spectrum mix of all orders because their pairwise independent joinings are necessarily product. Del Junco and Rudolph (1987) formalized the property ID(2,n)ID(2,n) (every pairwise-independent nn-fold self-joining is the product; an action satisfying it for all n>2n>2 is called PID, pairwise independently determined) and asked whether a zero-entropy mixing automorphism can fail ID(2,n)ID(2,n); their problem is a broader automorphism-level relative of the question below, listed as open in Ryzhikov's 2024 surveys.

De la Rue (2006) asked what a joining-based imitation of Ledrappier in one dimension would force, and proved (Theorem 3.5) that a 3-dot-type self-joining — pairwise independent, with ξi′′=f(ξi,ξi′)\xi''_i=f(\xi_i,\xi'_i) for all ii and some map f:A×A→Af:\mathbb{A}\times\mathbb{A}\to\mathbb{A} — implies ξ\xi is periodic or has entropy at least log⁡2\log 2, so no zero-entropy 2-fold mixing process carries one. The only known non-trivial pairwise-independent joinings occur for the uniform Bernoulli process (with ξ′′=ξ+ξ′\xi''=\xi+\xi' modulo nn) and for periodic processes, both excluded by zero entropy plus weak mixing (a process is weakly mixing when it mixes in the time-averaged sense, equivalently when its product with an independent copy is ergodic; 2-fold mixing implies weak mixing). Janvresse and de la Rue (2008) then treated the case where ξ0′′\xi''_0 depends on only finitely many coordinates of ξ\xi and ξ′\xi' (the PIJ property): they proved such a joining forces positive entropy unless the process is periodic, classified the irreducible aperiodic subshifts of finite type (a subshift of finite type is a closed shift-invariant set of sequences determined by finitely many forbidden blocks) admitting such joinings as measure-theoretically uniform Bernoulli, and conjectured no other PIJ examples exist. What no technique currently reaches is a joining with genuinely measurable coordinate dependence — ξ′′=φ(ξ,ξ′)\xi''=\varphi(\xi,\xi') almost surely for a Borel φ:AZ×AZ→AZ\varphi:\mathbb{A}^{\mathbb{Z}}\times\mathbb{A}^{\mathbb{Z}}\to\mathbb{A}^{\mathbb{Z}} possibly using infinitely many coordinates — or a non-product pairwise-independent joining with no functional dependence at all; the finite-width combinatorial and entropy arguments behind Theorem 3.5 and the PIJ theorem break down exactly there.

Problem Statement

Let ξ=(ξi)i∈Z\xi=(\xi_i)_{i\in\mathbb{Z}} be a stationary process on a finite alphabet A\mathbb{A} with distribution μ\mu, let λ\lambda be a 3-fold self-joining of ξ\xi (a stationary measure on (AZ)3(\mathbb{A}^{\mathbb{Z}})^3 whose three marginals equal μ\mu), and call λ\lambda pairwise independent if each of its two-dimensional marginals is μ⊗μ\mu\otimes\mu. Determine whether there exists a zero-entropy, 2-fold mixing stationary process ξ\xi admitting a pairwise-independent 3-fold self-joining λ≠μ⊗μ⊗μ\lambda\neq\mu\otimes\mu\otimes\mu (de la Rue 2006, Question 3.3; related to, but strictly narrower than, the automorphism-level del Junco–Rudolph problem on pairwise-independent joinings, ID(2,3)ID(2,3)). In particular, settle the structural variant emphasized by the source: if the coordinates of a pairwise-independent 3-fold self-joining satisfy ξ′′=φ(ξ,ξ′)\xi''=\varphi(\xi,\xi') λ\lambda-almost surely for some Borel measurable φ:AZ×AZ→AZ\varphi:\mathbb{A}^{\mathbb{Z}}\times\mathbb{A}^{\mathbb{Z}}\to\mathbb{A}^{\mathbb{Z}} that may depend on arbitrarily many coordinates — the case of φ\varphi depending on only finitely many coordinates being already resolved (periodic or positive entropy) — what structural consequences follow for ξ\xi; can it be 2-fold mixing with zero entropy? The general question ranges over all pairwise-independent 3-fold self-joinings, whereas the strictly narrower measurable-φ\varphi variant above restricts to those joinings that satisfy ξ′′=φ(ξ,ξ′)\xi''=\varphi(\xi,\xi').

References

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