Adelic Zariski dense orbit conjecture

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. Adelic Zariski Dense Orbit Conjecture

   Let $\mathbf{k}$ be an algebraically closed field of characteristic $0$ whose transcendence degree over $\mathbb{Q}$ is finite, and let $X$ be a variety over $\mathbf{k}$. Let $f\colon X\dashrightarrow X$ be a dominant rational map, and write $\mathbf{k}(X)^f=\{g\in\mathbf{k}(X):f^*g=g\}$. For $x\in X(\mathbf{k})$, write $O_f(x)=\{f^n(x)\mid n\geq 0\}$ when every iterate is defined. Adelic Zariski Dense Orbit Conjecture. If $\mathbf{k}(X)^f=\mathbf{k}$, then there exists a nonempty adelic open subset $A\subseteq X(\mathbf{k})$ such that for every $x\in A$, the $f$-orbit is well-defined and Zariski dense in $X$.

   This strengthens the Zariski Dense Orbit Conjecture by requiring every point in a nonempty adelic open set to have a well-defined Zariski dense orbit. Its status is not specified in the supplied text.

   source: Jia Jia, Takahiro Shibata, Junyi Xie and De-Qi Zhang, “Endomorphisms of quasi-projective varieties – towards Zariski dense orbit and Kawaguchi-Silverman conjectures”, arXiv:2104.05339 (2024).