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 ?. Then there exists a point x∈ X(F) with Zariski dense orbit under ? if and only if ? preserves no nontrivial rational fibration, i.e., there exists no nonconstant rational functionsf∈ F(X) such that ?*(f)=f. The Medvedev-Scanlon conjecture holds when F is uncountable. The case where F is countable (e.g., F=Qbar) is much more difficu herethe Medvedev-Scanlon conjecture has only been proved in a small number of special cases. In this paper we show thatthe 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.
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