We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $(\mathbb{C}^{2},0)$ there exist either an analytic curve of fixed points tangent to $[v]$ or $k$ parabolic manifolds where all the orbits are tangent to $[v]$, and that at least one of these parabolic manifolds is or contains a parabolic curve.
Consider a complex one dimensional foliation on a complex surface near a
singularity $p$. If $\mathcal{I}$ is a closed invariant set containing the
singularity $p$, then $\mathcal{I}$ contains either a separatrix at $p$ or an
invariant real three dimensional manifold singular at $p$
We prove that a C ∞ equivalence between germs holomorphic foliations at (C 2 , 0) establishes a bijection between the sets of formal separatrices preserving equisingularity classes. As a consequence, if one of the foliations is of second type, so is the other and they are equisingular.2000 Mathematics Subject Classification: 32S65. 2
We prove that the algebraic multiplicity of a holomorphic vector field at an isolated singularity is invariant by topological equivalences which are differentiable at the singular point.
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