Motivated by work of Zhang from the early `90s, Medvedev and Scanlon formulated the following conjecture. Let F be an algebraically closed field of characteristic 0 and let X be a quasiprojective variety defined over F endowed with a dominant rational self-map ϕ. Then there exists a point x∈ X(F) with Zariski dense orbit under ϕ if and only if ϕ preserves no nontrivial rational fibration, i.e., there exists no nonconstant rational functions f∈ F(X) such that ϕ*(f)=f. The Medvedev-Scanlon conjecture holds when F is uncountable. The case where F is countable (e.g., F=Qbar) is much more difficult; here the Medvedev-Scanlon conjecture has only been proved in a small number of special cases. In this paper we show that the Medvedev-Scanlon conjecture holds for all varieties of positive Kodaira dimension, and explore the case of Kodaira dimension 0. Our results are most definitive in dimension 3.

The authors have been partially supported by Discovery Grants from the National Science and Engineering Board of Canada.