Complete Conjugacy Invariants of Nonlinearizable Holomorphic Dynamics

Details Complete Conjugacy Invariants of Nonlinearizable Holomorphic Dynamics Complete Conjugacy Invariants of Nonlinearizable Holomorphic Dynamics

2009-03-13

arxiv.org/abs/0903.2394v1

Mathematics

Dynamical Systems

11 pages

Scientific paper

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$(\bf C,0)$. We also show that two nonlinearizable germs are conjugate if and only if their rotation numbers are equal and 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.

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