The Dynamical Manin-Mumford problem is a dynamical question inspired by classical results from arithmetic geometry.

In the setting of regular polynomial endomorphisms of C^2 of degree d>=2, it tasks to determine whether an algebraic curve containing infinitely many preperiodic points must be itself preperiodic.

In a work in collaboration with Romain Dujardin and Charles Favre, we prove this conclusion to hold, provided that: (★) the dynamics at infinity has no superattracting periodic points.

The proof is an interesting blend of techniques from arithmetic geometry and complex/non-archimedean dynamics.