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Complete conjugacy invariants of nonlinearizable holomorphic dynamics
Perez-Marco proved the existence of non-trivial totally invariant
connected compacts called hedgehogs near the fixed point of a
nonlinearizable germ of holomorphic diffeomorphism. We show that
if two nonlinearisable holomorphic germs with a common indifferent
fixed point have a common hedgehog then they must commute. This
allows us to establish a correspondence between hedgehogs and
nonlinearizable maximal abelian subgroups of Diff($\mathbb{C},0$).
We also show that two nonlinearizable germs with the same rotation
number are conjugate if and only if a hedgehog of one can be
mapped conformally onto a hedgehog of the other. Thus the
conjugacy class of a nonlinearizable germ is completely determined
by its rotation number and the conformal class of its hedgehogs.