Preprint C01/2001
Groups of germs of analytic diffeomorphisms
Fabio Enrique Brochero Martinez
Keywords: analytic germ | convergent orbits | formal and analytic conjugation
The study of germs of holomorphic diffeomorphisms and finitely generated groups of analytic germs of diffeomorphisms fixing the origin in one complex variable was started in the XIX century, and has been intensively studied by mathematicians in the past century. We deal with germs of diffeomorphisms in $(\C^2,0)$ and the finitely generated group of germs First, we study the finite groups of diffeomorphisms. We prove a generalization of Mattei-Moussu topological criteria about finiteness of a group.we compare a topological conjugacy class of some finite diffeomorphisms with their analytical conjugacy class, i.e. we construct the moduli space (topological vs analytical) of the diffeomorphisms that are conjugate with some finite order diffeomorphism. Next, we study the groups of diffeomorphisms supposing that they have some algebraic structure. We prove that if $\Cal G\subset \dihdos{}$ is a solvable group then its $7^{th}$ commutator subgroup is trivial. Furthermore, we characterize the abelian subgroup of diffeomorphisms tangent to the identity, and in the case when the group contains a dicritic diffeomorphism, i.e. the group contains a diffeomorphism $F(X)=X+F_{k+1}(X)+\cdots$ where $F_{k+1}(X)=f(X)X$ and $f$ is a homogeneus polynomial of degree $k$, we prove that the group is a subgroup of a one parameter group. We write $\dihdos{_1}$ to denote the group of diffeomorphisms tangent to the identity at $0\in \C^2$. We analyze the behavior of the orbits of a diffeomorphism tangent to the identity. We prove a generalization of the one dimensional flower theorem to two dimensional dicritic diffeomorphisms. Finally, we show the formal classification of diffeomorphisms tangent to the identity using the notion of the semiformal conjugacy. We show that a representative diffeomorphism found using semiformal conjugacy and a cocycle determine its formal conjugacy class.