Matteo Ruggiero - January 30, 2026 скачать в хорошем качестве

Matteo Ruggiero - January 30, 2026

Title: On the Dynamical Manin Mumford problem for polynomial endomorphisms of the plane Abstract: 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 bigger than 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. Condition (*) is crucial for our approach: it ensures that we can work near the Julia set at infinity at some place, and that the set W where orbits converge at super-exponential speed d at a fixed point at infinity is a (invariant) curve. If time allows, I will also present our recent results about the properties of W in the superattracting case.