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DCDS

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.

DCDS

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.

DCDS

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.

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