# American Institute of Mathematical Sciences

2010, 26(3): 847-856. doi: 10.3934/dcds.2010.26.847

## Complete conjugacy invariants of nonlinearizable holomorphic dynamics

 1 Ramakrishna Mission Vivekananda University, Belur Math, WB-711202, India

Received  May 2009 Revised  August 2009 Published  December 2009

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.
Citation: Kingshook Biswas. Complete conjugacy invariants of nonlinearizable holomorphic dynamics. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 847-856. doi: 10.3934/dcds.2010.26.847
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