The Zariski dense orbit conjecture

In this talk, I will present an adelic version of the Zariski dense orbit conjecture by Junyi Xie which is a strengthening of the original conjecture formulated independently by Medvedev-Scanlon, Amerik-Campana, and Zhang. I will introduce the adelic topology, which is a stronger topology than the Zariski topology, and its main properties. Using the adelic topology, one can give simpler proofs of the conjecture in the case of endomorphisms of abelian varieties and endomorphisms of A^n given by the coordinatewise action of one-variable polynomials. If time permits, we will discuss Xie’s proof of the conjecture in the case of the endomorphisms of P^2.