Joint ergodicity on nilpotent systems (Conjecture 5.5)

Prove or disprove that for any nilpotent system (X,𝔛,ΞΌ,T_1,...,T_β„“), the sequence (T_1^n,...,T_β„“^n)_n is jointly ergodic if and only if (i) T_1Γ—β‹―Γ—T_β„“ is ergodic on (X^β„“,𝔛^{βŠ—β„“},ΞΌ^β„“) and (ii) for all distinct i,j the group ⟨T_i^n T_j^{-n}: nβˆˆβ„€βŸ© acts ergodically on (X,𝔛,ΞΌ).

Background

Bergelson and Leibman proved the two-condition characterization for β„“=2.

Extending the characterization to arbitrary β„“ would parallel the commuting-ergodic case and solidify understanding of nilpotent actions.

References

Prove or disprove the following: for any nilpotent system $(X, , \mu,$! $T_1, \ldots, T_\ell)$, the action $(T_1n, \ldots, T_\elln)_n$ is jointly ergodic for $(X,,\mu)$ if and only if the following two conditions hold: \begin{enumerate} \item $T_1 \times \cdots \times T_\ell$ is ergodic on $(X\ell, {\otimes \ell}, \mu\ell)$; \item for any distinct $1\leq i, j\leq \ell$, the group $\langle T_inT_j{-n}\colon n\in\rangle$ acts ergodically on $(X, , \mu)$. \end{enumerate}

— Joint ergodicity - 40 years on  (2603.18974 - Kuca, 19 Mar 2026) in Section 7.4 (Joint ergodicity on nilpotent actions)