Complete conjugacy invariants of nonlinearizable holomorphic dynamics

Perez-Marco proved the existence of non-trivial totally invariant connectedcompacts called hedgehogs near the fixed point of a nonlinearizable germ ofholomorphic diffeomorphism. We show that if two nonlinearisable holomorphicgerms with a common indifferent fixed point have a common hedgehog then theymust commute. This allows us to establish a correspondence between hedgehogsand nonlinearizable maximal abelian subgroups of Diff$(\bf C,0)$. We also showthat two nonlinearizable germs are conjugate if and only if their rotationnumbers are equal and a hedgehog of one can be mapped conformally onto ahedgehog of the other. Thus the conjugacy class of a nonlinearizable germ iscompletely determined by its rotation number and the conformal class of itshedgehogs.