The polynomial Furstenberg joining and its applications

In this paper, a polynomial version of the Furstenberg joining is introduced and its structure is investigated. Particularly, it is shown that if all polynomials are non-linear, then almost every ergodic component of the joining is a direct product of an infinite-step pro-nilsystem and a Bernoulli system.As applications, some new convergence theorems are obtained. Particularly, it is proved thatif $T$ and $S$ are ergodic measure-preserving transformations on a probability space $(X,\X,\mu)$ and $T$ has zero entropy, then for all $c_i\in~\Z\setminus~\{0\}$, all integral polynomials $p_j$ with $\deg~{p_j}\geq~2$, and all $f_i,~g_j\in~L^\infty(X,\mu)$, $1\leq~i\leq~m$ and$1\leq~j\leq~d$, exists in $L^2(X,\mu)$,which extends a recent result by Frantzikinakis and Host (2023).Moreover, it is shown that for an ergodic measure-preserving system $(X,\X,\mu,T)$, a non-linear integral polynomial $p$ and $f\in~L^\infty(X,\mu)$, the Furstenberg systems of $~(f(T^{p(n)}x)~)_{n\in~\Z}$ are ergodic and isomorphic to direct products of infinite-step pro-nilsystems and Bernoulli systems for almost every $x\in~X$, which answers a problem by Frantzikinakis (2022).