Connectedness of polynomial diagonal orbit closures for minimal nilrotations and applications

Abstract

For a minimal nilrotation on a compact connected nilmanifold, we prove that the polynomial diagonal orbit closure associated with any finite family of polynomials with integer coefficients vanishing at the origin is connected. This resolves a conjecture of Glasscock, Koutsogiannis, Le, Moreira, Richter, and Robertson. Combined with their equivalence theorem, our result yields polynomial multiple recurrence in every prescribed residue class in topological dynamics, provided that the corresponding power of the transformation is minimal. Furthermore, we independently establish the measure-theoretic counterpart of this recurrence phenomenon. Finally, we construct a totally minimal nilsystem for which the lower central series identity proposed by Leibman fails.