We prove a topological rigidity theorem for transverse dynamics of real analytic foliations of codimension one (Theorem 1) as well as for pseudogroups of real analytic diffeomorphisms of open neighbourhoods of $0$ in the real line $\mathbb {R}$ (Theorem 3). We apply those results to prove the topological rigidity of analytic actions of the surface group $\Gamma ^g$ on the circle $S^1$ (Corollary 4) and also the topological invariance of the Godbillon-Vey (Corollary 2).