Donoso–Koutsogiannis–Sun joint ergodicity conjecture for polynomial iterates

Let $(X,\mathcal X,\mu)$ be a probability space, let $T_1,\ldots,T_\ell$ be commuting measure-preserving transformations, and let $p_1,\ldots,p_\ell\in\mathbb Z[n]$. A sequence of commuting transformations $(S_n)_{n\in\mathbb N}$ is ergodic for a measure if

$$\lim_{N\to\infty}\left\Vert\frac{1}{N}\sum_{n=1}^N S_n f-\int f\,d\mu\right\Vert_{L^2(\mu)}=0$$

for every $f\in L^\infty(\mu)$. The polynomials $p_1,\ldots,p_\ell$ are jointly ergodic for $(X,\mathcal X,\mu,T_1,\ldots,T_\ell)$ if and only if the following two conditions are satisfied: for all distinct $i,j\in\{1,\ldots,\ell\}$, $(T_i^{p_i(n)}T_j^{-p_j(n)})_{n\in\mathbb N}$ is ergodic for $\mu$, and $(T_1^{p_1(n)}\times\cdots\times T_\ell^{p_\ell(n)})_{n\in\mathbb N}$ is ergodic for $\mu\times\cdots\times\mu$. This criterion characterizes joint ergodicity of polynomial iterates and is presented as a special case of the conjecture of Donoso, Koutsogiannis, and Sun; in the source paper, the stated special case is subsequently proved as a corollary of the authors’ stronger theorem.