On the Zariski dense orbit conjecture

Abstract: 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.

dynamical systems

Audience: researchers in the topic

BIRS workshop: Algebraic Dynamics and its Connections to Difference and Differential Equations

Series comments: The field of algebraic dynamics has emerged over the past two decades at the confluence of algebraic geometry, discrete dynamical systems, and diophantine geometry. In recent work, striking connections have been observed between algebraic dynamics and much older theories of difference and differential equations. This meeting brings together mathematicians with expertise in such diverse fields as ring theory, complex dynamics, differential and difference algebra, combinatorics and algebraic geometry. New work towards the dynamical Mordell-Lang and dense orbit conjectures as well as theorems on hypertranscendence and functional independence proven by connecting difference Galois theory, algebraic dynamics and other algebraic approaches to the study offunctional equations will be presented at this meeting.