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In this paper we prove the following theorem. Let 
$f$ be a dominant polynomial endomorphism of the affine plane defined over an algebraically closed field of characteristic 
$0$. If there is no nonconstant invariant rational function under 
$f$, then there exists a closed point in the plane whose orbit under 
$f$ is Zariski dense. This result gives us a positive answer to a conjecture proposed by Medvedev and Scanlon, by Amerik, Bogomolov and Rovinsky, and by Zhang for polynomial endomorphisms of the affine plane.
