Math Conjectures

Rokhlin multiple-mixing problem

OPENMajorOpen problemProposed 1949 · Full conjecture

Canonical statement

Let TT be an invertible measure-preserving transformation of a standard probability space (X,B,μ)(X,\mathcal B,\mu). If μ(A∩T−nB)→μ(A)μ(B)\mu(A\cap T^{-n}B)\to\mu(A)\mu(B) as n→∞n\to\infty for all A,B∈BA,B\in\mathcal B, then for every k≥2k\ge2, all A0,…,Ak∈BA_0,\ldots,A_k\in\mathcal B, and all integer sequences n1(j),…,nk(j)n_1^{(j)},\ldots,n_k^{(j)} satisfying, as j→∞j\to\infty,
min⁡i∣ni(j)∣→∞,min⁡i≠l∣ni(j)−nl(j)∣→∞, \min_i|n_i^{(j)}|\to\infty,\qquad \min_{i\ne l}|n_i^{(j)}-n_l^{(j)}|\to\infty,
one has
μ ⁣(A0∩⋂i=1kT−ni(j)Ai)⟶∏i=0kμ(Ai). \mu\!\left(A_0\cap\bigcap_{i=1}^kT^{-n_i^{(j)}}A_i\right) \longrightarrow\prod_{i=0}^k\mu(A_i).
View source LaTeX
Let \(T\) be an invertible measure-preserving transformation of a standard probability space \((X,\mathcal B,\mu)\). If \(\mu(A\cap T^{-n}B)\to\mu(A)\mu(B)\) as \(n\to\infty\) for all \(A,B\in\mathcal B\), then for every \(k\ge2\), all \(A_0,\ldots,A_k\in\mathcal B\), and all integer sequences \(n_1^{(j)},\ldots,n_k^{(j)}\) satisfying, as \(j\to\infty\), \[ \min_i|n_i^{(j)}|\to\infty,\qquad \min_{i\ne l}|n_i^{(j)}-n_l^{(j)}|\to\infty, \] one has \[ \mu\!\left(A_0\cap\bigcap_{i=1}^kT^{-n_i^{(j)}}A_i\right) \longrightarrow\prod_{i=0}^k\mu(A_i). \]
DYN-010:[Rokhlin1949MultipleMixing][Ledrappier1978MixingNotThree][Ryzhikov2024MultipleMixing75][Ryzhikov2024JoiningsProblems]
Dynamical systemsProbability theory

Notes

Rokhlin's problem, posed in 1949, asks whether ordinary (two-fold) mixing already forces mixing of every order: if μ(A∩T−nB)→μ(A)μ(B)\mu(A\cap T^{-n}B)\to\mu(A)\mu(B) for all measurable sets, must μ(A0∩T−n1A1∩⋯∩T−nkAk)→∏iμ(Ai)\mu(A_0\cap T^{-n_1}A_1\cap\cdots\cap T^{-n_k}A_k)\to\prod_i\mu(A_i) whenever the times and all their mutual differences tend to infinity? The question originates in Rokhlin's study of endomorphisms of compact commutative groups [Rokhlin1949MultipleMixing].

Affirmative answers are known for large classes of systems: many rank-one transformations, algebraic and Gaussian systems, and systems satisfying finite-entropy hypotheses are mixing of all orders. The restriction to a single Z\mathbb Z-action is essential: Ledrappier constructed a mixing Z2\mathbb Z^2-action of zero entropy that fails three-fold mixing [Ledrappier1978MixingNotThree], so the naive higher-rank analogue is false. Surveys of the state of the problem, seventy-five years on, are given by Ryzhikov [Ryzhikov2024MultipleMixing75] [Ryzhikov2024JoiningsProblems].

No mixing Z\mathbb Z-action failing three-fold mixing is known, and no proof that two-fold mixing implies three-fold mixing exists either; resolving the problem requires producing one or the other, and it remains open.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.