I asked this question on Math Stack Exchange at https://math.stackexchange.com/questions/4739742/; it received 4 upvotes, but no comments or answers even after a 450-point bounty.
The question:
Is it known whether, for every mixing measure-preserving dynamical system $(X,\mathcal{B}(X),\mu,T)$ with $X$ a compact metrisable space and $T$ a continuous map, we have that for all $A,B,C \in \mathcal{B}(X)$, $$ \mu(A \cap T^{-k}(B) \cap T^{-(k+l)}(C)) - \mu(A \cap T^{-k}(B) \cap T^{-(k+l+1)}(C)) \to 0 \quad \text{as } (k,l) \to (\infty,\infty) ? $$
[Well, I guess it can't be the case that the answer is known to be no, as that would then solve Rokhlin's multiple mixing problem. But perhaps it's known (or not too hard to show) that the answer is yes?]
I ask because:
- if we were to have a counterexamplary $(X,\mathcal{B}(X),\mu,T)$, then (unless I am mistaken) it is not hard to show that one can find $A,B \in \mathcal{B}(X)$ with $\mu(A),\mu(B)>0$, sequences $k_n,l_n \to \infty$, and a non-$T$-invariant probability measure $\nu$ on $X$, such that for all $C \in \mathcal{B}(X)$, $$ \mu(A \cap T^{-k_n}(B) \cap T^{-(k_n+l_n)}(C)) \to \mu(A)\mu(B)\nu(C) \quad \text{as } n \to \infty; $$
- but (again, unless I am mistaken) it is not hard to show that in general, for any mixing measure-preserving dynamical system $(X,\mathcal{X},\mu,T)$, for any $A,B,C \in \mathcal{X}$ with $\mu(A)=\mu(B^c)$, $$ \sup_{k \geq 0} \Big( \mu(A \cap T^{-k}(B) \cap T^{-(k+l)}(C)) - \mu(A^c \cap T^{-k}(B^c) \cap T^{-(k+l)}(C)) \Big) \to 0 \quad \text{as } l \to \infty; $$
and so, if in the first bullet point we can take $A,B$ with $\mu(A)=\mu(B^c)$, then writing \begin{align*} \mu_k &:= \mu(\,\cdot\,|A \cap T^{-k}(B)) \\ \tilde{\mu}_k &:= \mu(\,\cdot\,|A^c \cap T^{-k}(B^c)), \end{align*} we have that for all sufficiently large $n$,
- on the one hand, $\,\mu_{k_n} \!\approx \mu(\,\cdot\,|A)\,$ and $\,\tilde{\mu}_{k_n} \!\approx \mu(\,\cdot\,|A^c)$;
- and yet on the other hand, the trajectories of $\mu_{k_n}$ and $\tilde{\mu}_{k_n}$ under pushforwards of $T$ approximately meet at the same non-invariant measure $\nu$ before approaching $\mu$.
This kind of behaviour feels like quite an awkward behaviour to try and construct for an autonomous dynamical system (in contrast to a non-autonomous system where you're allowed to change the map at each step so as to "brute-force" whatever qualitative behaviour you like).
