  In this paper we prove the following theorem. Let $f:\mathbb{A}^2\rightarrow
\mathbb{A}^2$ be a dominate polynomial endomorphisms defined over an
algebraically closed field $k$ of characteristic $0$. If there are no
nonconstant rational function $g:\mathbb{A}^2-rightarrow \mathbb{P}^1$
satisfying $g\circ f=g$, then there exists a point $p\in \mathbb{A}^2(k)$ whose
orbit under $f$ is Zariski dense in $\mathbb{A}^2$.
  This result gives us a positive answer to a conjecture of Amerik, Bogomolov
and Rovinsky ( and Zhang) for polynomial endomorphisms on the affine plane.
