Rapidly convergent iteration method for simultaneous normal forms of commuting maps
We propose a new rapidly convergent KAM type iteration scheme under weak simultaneous arithmetic conditions for overdetermined systems related to simulataneous conjugation of commuting maps. The first application is for the simultaneous linearization of commuting holomorphic maps \(\phi(x),\dots, \Phi_d(x)\) in \(\mathbb{C}^n\) fixing the origin, not necessarily diffeomorphisms if \(n\geq 2\), \(d\geq 2\). The second one is for simultaneous local reduction of orientation preserving Gevrey \(G^\sigma\) \((\sigma\geq 1)\) circle mappings. Here one of the crucial ingredients of the proof is to show some new nonlinear superposition estimates in scales of Banach spaces of Gevrey functions.
scientific article; zbMATH DE number 1336873