Zariski Dense Orbit conjecture for rational self-maps

Let $X$ be a smooth projective variety over $\overline{{\mathbb Q}}$, let $f$ be a dominant rational self-map of infinite order, and let

$$\mathcal{Z}(f):=\{x\in X_f(\overline{{\mathbb Q}})\mid \operatorname{Orb}_f(x)\text{ is Zariski dense in }X\}.$$

Zariski Dense Orbit conjecture. If there is no non-constant rational function $g$ on $X$ satisfying $g\circ f=g$, then $\mathcal{Z}(f)\neq\emptyset$.

This conjecture gives a geometric criterion for the existence of Zariski-dense algebraic orbits. The paper introduces it as directly related to the existence question for such orbits; the general statement remains open, although affirmative results are known in several cases, including certain birational surface maps.