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.
Abstract
We deal with complex points of two-dimensional surfaces. A short review of basic results about complex points of smooth surfaces in is presented at the beginning. For algebraic surfaces, a formula is proved which expresses the number of complex points as the local degree of an explicitly constructible polynomial endomorphism.
The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane
