Medvedev–Scanlon–Amerik–Campana Zariski dense orbit conjecture

Let $X$ be a quasiprojective variety defined over an algebraically closed field $K$ of characteristic $0$, and let $\Phi:X\dashrightarrow X$ be a dominant rational self-map. An orbit is well-defined if every iterate avoids the indeterminacy locus of $\Phi$.

Zariski dense orbit conjecture. Either there exists $\alpha\in X(K)$ whose orbit under $\Phi$ is well-defined and Zariski dense in $X$, or there exists a non-constant rational function $f:X\dashrightarrow {\mathbb P}^1$ such that $f\circ\Phi=f$.

This conjecture was formulated independently by Medvedev and Scanlon and by Amerik and Campana, following a question of Zhang. Several partial results are known, but the conjecture in this generality remains open.