Divergent diagrams of folds and simultaneous conjugacy of involutions (Q1779045)

A fold is a germ \(f\) at the origin of a self-map of \({\mathbb R}^n\) such that \(k^{-1}\circ f\circ h(x)=(x_1^2,x_2,\ldots,x_n)\) for suitable germs \(h\), \(k\) of diffeomorphisms of \({\mathbb R}^n\) fixing the origin. A divergent diagram of folds of rank \(s\) is just an \(s\)-tuple of folds of the same \({\mathbb R}^n\). Two divergent diagrams of folds \((f_1,\ldots,f_s)\) and \((g_1,\ldots,g_s)\) are equivalent if there exist germs \(h,k_1,\ldots,k_s\) of diffeomorphisms such that \(g_j=k_j\circ f_j\circ h^{-1}\) for \(j=1,\ldots,s\). To any fold \(f\) it is possible to associate a unique involution \(\phi\neq\text{ Id}\) of \({\mathbb R}^n\) such that \(f\circ\phi=f\). Analogously, to any divergent diagram of folds of rank \(s\) one can associate a unique \(s\)-tuple of involutions, and the authors prove that two divergent diagrams of folds are equivalent if and only if the associated \(s\)-tuple of involutions are. Using these results, they give normal forms for divergent diagrams of folds (and the associated involutions) of rank 2.