Medvedev–Scanlon Zariski dense orbit conjecture

Let $X$ be a projective variety over an algebraically closed field $k$ of characteristic zero, and let $f:X\dasharrow X$ be a dominant rational self-map. Write $k(X)^f$ for the field of rational functions invariant under $f$, let $X_f(k)$ denote the points whose forward $f$-orbits are defined, and write $\mathcal O_f(x)$ for the orbit of $x$. Zariski dense orbit conjecture. Either

$$k(X)^f\ne k,$$

or there is a point $x\in X_f(k)$ whose orbit $\mathcal O_f(x)$ is Zariski dense in $X(k)$. This conjecture links invariant rational functions with the existence of dense dynamical orbits; the source presents it as a conjecture proposed by Medvedev and Scanlon, Amerik, Bogomolov and Rovinsky, strengthening a conjecture of S.-W. Zhang. The source does not establish the conjecture in full generality, although it proves cases for automorphisms of projective varieties with irregularity at least their dimension minus one and obtains a reduced result for projective threefolds.