Complete conjugacy invariants of nonlinearizable holomorphic dynamics
Kingshook Biswas
Discrete & Continuous Dynamical Systems - A 2010, 26(3): 847-856 doi: 10.3934/dcds.2010.26.847
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.
keywords: nonlinearizable dynamics hedgehogs. Irrationally indifferent fixed points
Smooth combs inside hedgehogs
Kingshook Biswas
Discrete & Continuous Dynamical Systems - A 2005, 12(5): 853-880 doi: 10.3934/dcds.2005.12.853
We use techniques of tube-log Riemann surfaces due to R.Pérez-Marco to construct a hedgehog containing smooth $C^\infty$ combs. The hedgehog is a common hedgehog for a family of commuting non-linearisable holomorphic maps with a common indifferent fixed point. The comb is made up of smooth curves, and is transversally bi-Hölder regular.
keywords: combs. hedgehogs Irrational indifferent fixed points
Maximal abelian torsion subgroups of Diff( C,0)
Kingshook Biswas
Discrete & Continuous Dynamical Systems - A 2011, 29(3): 839-844 doi: 10.3934/dcds.2011.29.839
In the study of the local dynamics of a germ of diffeomorphism fixing the origin in $\mathbb C$, an important problem is to determine the centralizer of the germ in the group Diff$(\mathbb C,0)$ of germs of diffeomorphisms fixing the origin. When the germ is not of finite order, then the centralizer is abelian, and hence a maximal abelian subgroup of Diff$(\mathbb C,0)$. Conversely any maximal abelian subgroup which contains an element of infinite order is equal to the centralizer of that element. A natural question is whether every maximal abelian subgroup contains an element of infinite order, or whether there exist maximal abelian torsion subgroups; we show that such subgroups do indeed exist, and moreover that any infinite subgroup of the rationals modulo the integers $\mathbb{Q/Z}$ can be embedded into Diff$(\mathbb C,0)$ as such a subgroup.
keywords: Germs of holomorphic diffeomorphisms centralizers.

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