From the Perspective of Nonsolvable Dynamics on \((\mathbb C,0)\) : Basics and Applications

The structure of pseudo groups of holomorphic diffeomorphisms on \(\mathbb C\) with a fixed point at the origin has been intensively studied from various view points for the last few decades. We recall the fundamental results on this subject beginning from the basics, and introduce some recent developments. We provide also a novel proof of the existence of fixed points by “closing an orbit by collision”, and discuss the topological rigidity from a new perspective with the \(\lambda \) -lemma and the quasi conformal mapping theory, and we answer partially to a question posed by Rebelo and Reis [37] and Shcherbakov [43], proving that the linear multipliers of fixed points of a nonsolvable pseudogroup acting on \((\mathbb C,0)\) are dense in \(\mathbb C^*\) . In the last two sections we discuss the classification problem of implicit functions and yet another application to the problem of topological moduli of polynomial maps of \(\mathbb C^3\) to \(\mathbb C^2\) .