Polynomial Furstenberg joinings and its applications

In this paper, a polynomial version of 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 infinity-step pro-nilsystem and a Bernoulli system. As applications, some new convergence theorems are obtained. Particularly, it is proved that if $T$ and $S$ are ergodic measure preserving transformations on a probability space $(X,{\mathcal X},\mu)$ and $T$ has zero entropy, then for all $c_i\in {\mathbb Z}\setminus \{0\}$, all integral polynomials $p_j$ with $\deg {p_j}\ge 2$, and for all $f_i, g_j\in L^\infty(X,\mu)$, $1\le i\le m$ and $1\le j\le d$, $$\lim_{N\to\infty} \frac{1}{N}\sum_{n=0}^{N-1}f_1(T^{c_1n}x)\cdots f_m(T^{c_mn}x)\cdot g_1(S^{p_1(n)}x)\cdots g_d(S^{p_d(n)}x),$$ exists in $L^2(X,\mu)$, which extends the recent result by Host and Frantzikinakis. Moreover, it is shown that for an ergodic measure-preserving system $(X,{\mathcal X},\mu,T)$, a non-linear integral polynomial $p$ and $f\in L^\infty(X,\mu)$, the Furstenberg systems of $\big(f(T^{p(n)})x\big)_{n\in {\mathbb Z}}$ 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.