Rokhlin's multiple-mixing problem for a single transformation

Theorem 1.1: every invertible mixing probability-preserving transformation, on an arbitrary probability space with no standardness assumption, is mixing of every finite order. The proof assumes a least failing order $k$, reduces to a zero-entropy finite-alphabet process, builds saturated (sated) array extensions of the limit joinings, extracts orthogonal lines of modulus-one functions from row and column conditional-expectation operators, and cancels against them. Corollary: the same for mixing endomorphisms and for mixing flows whose Koopman operators are strongly continuous on $L^1$. Not shown: anything for $\mathbb Z^d$ or other group actions (where Ledrappier's example shows the analogue fails), or any rate of convergence.

Theorem 1.1: every invertible mixing probability-preserving transformation, on an arbitrary probability space with no standardness assumption, is mixing of every finite order. The proof assumes a least failing order $k$, reduces to a zero-entropy finite-alphabet process, builds saturated (sated) array extensions of the limit joinings, extracts orthogonal lines of modulus-one functions from row and column conditional-expectation operators, and cancels against them. Corollary: the same for mixing endomorphisms and for mixing flows whose Koopman operators are strongly continuous on $L^1$. Not shown: anything for $\mathbb Z^d$ or other group actions (where Ledrappier's example shows the analogue fails), or any rate of convergence.