Classification for joint ergodicity over ℤ^ℓ-polynomial actions

Prove or disprove that for polynomials a_1,...,a_ℓ∈ℤ^ℓ[t] and a system (X,𝔛,μ,T_1,...,T_ℓ), the sequence (T^{a_1(n)},...,T^{a_ℓ(n)})_n is jointly ergodic if and only if (i) (T^{a_1(n)}×⋯×T^{a_ℓ(n)})_n is ergodic on (X^ℓ,𝔛^{⊗ℓ},μ^ℓ) and (ii) (T^{a_i(n)}T_j^{-a_j(n)})_n is ergodic on (X,𝔛,μ) for all i≠j.

Background

The survey proves a complete classification for integer polynomials in one variable and poses the higher-dimensional analogue over ℤℓ.

This generalization would unify product and difference ergodicity criteria with joint ergodicity for polynomial actions of ℤℓ.

References

For polynomials in $\ell[t]$, the following more general problem remains open.

— Joint ergodicity - 40 years on  (2603.18974 - Kuca, 19 Mar 2026) in Section 4.1 (Joint ergodicity classification problem)

We expect that the distinct-degree assumption can be replaced by the weaker assumption of pairwise independence. More precisely, we propose the following conjecture, which may be viewed as a topological analogue of {Theorem 2.16. Let $(X,S_1,\dots,S_k)$ be a minimal $Zk$-system, let $T_1,\dots,T_d\in \langle S_1,\dots,S_k\rangle$ be transitive transformations, and let $p_1,\dots,p_d$ be pairwise independent integer polynomials. Then $(T_1{p_1(n)}){n},\dots,(T_d{p_d(n)}){n}$ are jointly transitive if and only if $(T_1{p_1(n)}\times \dots\times T_{d}{p_{d}(n)})_{n}$ is transitive on $Xd$.

— A saturation theorem for distinct-degree polynomials and an application to joint transitivity  (2608.26570 - Álvarez, 27 Aug 2026) in Conjecture in Section 1, subsection 'The joint transitivity problem'