Abstract. 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.