On the dynamic of holomorphic diffeomorphisms groups fixing a commune point, on C^n

TL;DR

This paper investigates the dynamics of groups of holomorphic diffeomorphisms fixing the origin in C^n, establishing conditions for orbit isomorphisms and properties of dense and minimal orbits, especially for abelian groups.

Contribution

It provides new conditions for orbit isomorphism to linearized orbits and demonstrates the existence of dense and minimal orbits for abelian groups fixing 0 in C^n.

Findings

Orbit G(x) is isomorphic to linear orbit L_G(x) under certain conditions.

Existence of a G-invariant dense open set U with relatively minimal orbits for abelian G.

If G has a dense orbit, then all orbits in U are dense in C^n.

Abstract

In this paper, we study the action on C^n of any group G of holomorphic diffeomorphisms (automorphisms) of C^n fixing 0. Suppose that there is x in C^n, having an orbit which generates C^n and also E(x)=C^n, where E(x) is the vector space generated by L_{G}={D_{0}fx, f in G }. We give an important condition so that an orbit G(x) is isomorphic (by linear map) to the orbit L_{G}(x)of the linear group L_{G}. More if G is abelian, we prove the existence of a G-invariant open set U, dense in C^n, in which every orbit O is relatively minimal (i.e. the closure of O in U is a closed non empty, G-invariant set and has no proper subset with these properties). Moreover, if G has a dense orbit in C^n then every orbit of U is dense in C^n.