Local rigidity of self-joinings and factors of pro-nilsystems

Abstract

It is an immediate consequence of the ergodic structure theorem of Host and Kra that every factor of an ergodic $k$-step pro-nilsystem is again an ergodic $k$-step pro-nilsystem. It has remained open whether this fact can be proved independently of the structure theorem itself. In this note, we give such a proof. The key new ingredient is a local rigidity theorem for nilsystems: any ergodic self-joining sufficiently close to the diagonal joining is necessarily the graph joining of an automorphism. This rigidity result may be of independent interest. As an application, our proof of the factor-closure of pro-nilsystems combined with a result of Tao yields a new proof of the ergodic structure theorem of Host and Kra from the combinatorial inverse theorem of Green, Tao, and Ziegler for the Gowers norms on cyclic groups. We can also use our methods to establish an independent proof of the factor-closure property of topological pro-nilsystems, a fact that can also be derived from the topological structure theorem of Host, Kra, and Maass.