Factor-Closure of Pro-Nilsystems via Local Rigidity of Nilsystems

Abstract

Pro-nilsystems arise naturally in ergodic theory through the study of multiple ergodic averages. The Host-Kra structure theorem implies that every factor of an ergodic $s$-step pro-nilsystem is again an $s$-step pro-nilsystem. No proof of this fact independent of the structure theorem was previously known. In this thesis, we give such a proof, using only arguments intrinsic to nilsystems. The main new ingredient is a local rigidity result for self-joinings of nilsystems: every ergodic self-joining sufficiently close to the diagonal joining is necessarily the graph joining of an automorphism. We prove this geometrically by showing that nilmanifolds cannot contain arbitrarily small nontrivial subnilmanifolds. We also obtain a direct proof of the analogous factor-closure result for transitive topological pro-nilsystems, previously known as a consequence of the Host-Kra-Maass topological structure theorem. Finally, we give a detailed account of Tao's deduction of a weak structure theorem directly from the Green-Tao-Ziegler inverse theorem for the Gowers norms. Combined with the factor-closure of pro-nilsystems, this yields a new proof of the Host-Kra structure theorem. The principal new results of this thesis also appear in a joint paper with Asgar Jamneshan (arXiv:2604.02089). This thesis provides a more self-contained treatment, including the necessary background on dynamical systems and nilsystems, as well as full details of Tao's argument.