The existence of Zariski dense orbits for endomorphisms of projective surfaces

Abstract:

Let $f$ be a dominant endomorphism of a smooth projective surface $X$ over an algebraically closed field $\mathbf {k}$ of characteristic $0$. We prove that if there is no rational function $H \in \mathbf {k}(X)$ such that $H \circ f = H$, then there exists a point $x \in X(\mathbf {k})$ such that the forward orbit of $x$ under $f$ is Zariski dense in $X$. This result gives us a positive answer to the Zariski dense orbit conjecture for endomorphisms of smooth projective surfaces.

We also define a new topology on varieties over algebraically closed fields with finite transcendence degree over $\mathbb {Q}$, which we call “the adelic topology”. This 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 version of the Zariski dense orbit conjecture that is stronger than the original one and quantifies how many such orbits there are. We also prove this adelic version for endomorphisms of smooth projective surfaces, for endomorphisms of abelian varieties, and for split polynomial maps. This yields new proofs of the original conjecture in the latter two cases.

In Appendix A, we study endomorphisms of $k$-affinoid spaces. We show that for certain endomorphisms $f$ on a $k$-affinoid space $X$, the attractor $Y$ of $f$ is a Zariski closed subset and that the dynamics of $f$ is semi-conjugate to its restriction to $Y$. 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 $(\mathbb {P}^1)^N$.