  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.
