Convergence of cataclysm deformations on Anosov representations and applications

Abstract

A cataclysm deformation, that shears and twists a given Anosov representation according to data known as a twisted transverse cocycle, is an intuitive and powerful tool for studying Anosov representations. We show that if a sequence of twisted measured laminations converges weakly, the sequence of corresponding cataclysm deformations on the space of Anosov representations converges uniformly on compact sets. This result leads to two applications. First, we obtain an extension of the Goldman product formula. Second, we consider strongly dense representations, introduced by Breuillard--Green--Guralnick--Tao and Long--Reid. Using cataclysm deformations, we show that, for a split real form $\mathsf{G}$ whose Weyl group contains $-1$, the set of strongly dense $\mathsf{G}$-Hitchin representations is not open in the $\mathsf{G}$-Hitchin component.