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On the Zariski dense orbit conjecture Xie, Junyi


We prove the following theorem. Let f be a dominant endomorphism of a 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 for endomorphisms of projective surfaces. We define a new canonical topology on varieties over an algebraically closed field which has finite transcendence degree over Q. We call it 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, which is stronger than the original one and quantifies how many such orbits there are. We also prove this adelic version for endomorphisms of projective surfaces, for endomorphisms of abelian varieties, and split polynomial maps. This yields new proofs of the original conjecture in the latter two cases.

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