The existence of Zariski dense orbits for endomorphisms of projective surfaces (with an appendix in collaboration with Thomas Tucker)

Abstract

In this paper we prove the following theorem. Let $f$ be a dominant endomorphism of a smooth projective surface over an algebraically closed field of characteristic $0$. If there is no nonconstant invariant rational function under $f$, then there exists a closed point whose orbit under $f$ is Zariski dense. This result gives us a positive answer to the Zariski dense orbit conjecture proposed by Medvedev and Scanlon, by Amerik, Bogomolov and Rovinsky, and by Zhang, for endomorphisms of smooth projective surfaces. Moreover, we define a new canonical topology on varieties over an algebraically closed field which has finite transcendence degree over $\mathbb{Q}$. We call it the adelic topology. The adelic topology is stronger than the Zariski topology and an irreducible variety is still irreducible in this topology. Using the adelic topology, we propose an adelic verison of the Zariski dense orbit conjecture. This version is stronger then the original one and it quantifies how many such orbits there are. We also proved this adelic version for endomorphisms of smooth projective surfaces. Moreover, we proved the adelic verison of the Zariski dense orbit conjecture for endomorphisms of abelian varieties and split polynomial maps. This yields new proofs for the original version in this two cases. In Appendix A, we study the endomorphisms on the $k$-affinoid spaces. We show that for certain endomorphism $f$ on a $k$-affinoid space $X$, the attractor $Y$ of $f$ is a Zariski closed subset and the dynamics of $f$ semi-conjugates to its restriction on $Y.$ A special case of this result is used in the proof of the main theorem. In Appendix B, written in collaboration with Thomas Tucker, we prove the Zariski dense orbit conjecture for endomorphisms of $(\mathbb{P}^1)^N.$