The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane

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.