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Motivated by work of Zhang from the early `90s, Medvedev and Scanlonformulated the following conjecture. Let $K$ be an algebraically closed fieldof characteristic $0$ and let $X$ be a quasiprojective variety defined over $K$endowed with a dominant rational self-map $\Phi$. Then there exists a point$\alpha\in X(K)$ with Zariski dense orbit under $\Phi$ if and only if $\Phi$preserves no nontrivial rational fibration, i.e., there exists no non-constantrational function $f\in K(X)$ such that $\Phi^*(f)=f$. The Medvedev-Scanlonconjecture holds when $K$ is uncountable. The case where $K$ is countable(e.g., $K=\overline{\mathbb{Q}}$) is much more difficult; here the conjecturehas only been proved in a small number of special cases. In this paper we showthat the Medvedev-Scanlon conjecture holds for all varieties of positiveKodaira dimension, and explore the case of Kodaira dimension $0$. Our resultsare most complete in dimension $3$.

On the Medvedev-Scanlon Conjecture for Minimal Threefolds of Non-Negative Kodaira Dimension