Orbit classification conjecture for points on K3 surfaces

Let $X$ be a complex K3 surface, let $\operatorname{Aut}(X)$ be its automorphism group, and let $\Gamma\subseteq \operatorname{Aut}(X)$ be a nonelementary subgroup, meaning that it contains a nonabelian free group. Let $\mu$ be a probability measure on a generating set of $\Gamma$. A real $2$-dimensional submanifold $S\subset X$ is totally real if, for every $s\in S$, the complex span of $T_sS$ agrees with $T_sX$. For any $x\in X$, the orbit classification conjecture. exactly one of the following possibilities occurs: (1) $\Gamma\cdot x$ is finite; (2) $\Gamma\cdot x$ is contained and Zariski-dense in an algebraic curve $C$; (3) there exists a $\Gamma$-invariant, $2$-dimensional set $S\subset X$ containing $x$, stratified by manifolds, whose open stratum $S^{\circ}$ is totally real and in which $\Gamma\cdot x$ is dense, with associated stationary measure a smooth volume on $S^{\circ}$; or (4) $\Gamma\cdot x$ is dense in $X$, with associated stationary measure the smooth volume on $X$. Every ergodic stationary measure is obtained as an empirical measure from a point as above, and, except for an invariant algebraic curve, the stationary measure is also invariant. The first case gives an atomic measure; in the second case, $C$ is isomorphic to a finite union of copies of $\mathbb{P}^1(\mathbb{C})$, the action on each component is by a subgroup of $\operatorname{PGL}_2(\mathbb{C})$, and the stationary measure is determined by this linear action through Furstenberg's theory. This conjecture organizes the possible orbit closures and stationary measures for nonelementary automorphism groups of K3 surfaces. The parabolic case is established by Cantat--Dujardin, while the totally real case remains to be understood in full generality; points on totally real surfaces without dense orbit are predicted to have either finite orbits or orbits confined to real algebraic curves.