Separatrizes de germes de campos de vetores analíticos reais em dimensão dois

Abstract

This thesis is dedicated to the study of conditions to ensure the existence of formal separatrices for a foliation defined by a germ of real analytical vector field with an algebraically isolated singularity at the origin of $\mathbb{R}^2$. We also present sufficient conditions to guarantee the existence of separatrices for germs of foliations defined by real analytical 1-forms at $(\mathbb{R}^3,0)$. At $(\mathbb{R}^2,0)$ we study the general case and at $(\mathbb{R}^3,0) $ we study only the non-dicritical case. In dimension two, a formal separatrix, or simply separatrix, is a germ of invariant irreducible formal curve, whereas in dimension three it is a germ of invariant irreducible formal surface. At $(\mathbb{R}^2,0)$, a germ of foliation $\mathcal{F}_\mathbb{R}$ induced by a germ of real analytical vector field with algebraically isolated singularity at the origin does not always admit formal separatrix. After a process of reduction of singularities, each singularity of saddle-node type obtained can be classified as topological saddle, topological node or topological saddle-node. We say that $\mathcal{F}_\mathbb{R}$ is of topological real generalized curve type if after a process of reduction of singularities it does not admit singularities of topological saddle-node type. Our main result is that \emph{if either the algebraic multiplicity or the Milnor number of a germ of a topological real generalized curve type foliation at $ (\mathbb{R}^2,0) $ is even, then it has at least one formal separatrix.} At $(\mathbb{R}^3,0)$, a germ of foliation $ \mathcal{F}_\mathbb{R}$ induced by a germ of integrable real analytical 1-form of codimension one is $\mathbb{C}$-non-dicritical if its complexification is a non-dicritical germ of holomorphic 1-form. A \emph{real immersion} $i_\mathbb{R}: (\mathbb{R}^2,0) \hookrightarrow (\mathbb{R}^3,0)$ is \emph{transversal} to $\mathcal{F}_\mathbb{R}$, if the singular set ${\rm Sing} (i_\mathbb{R}^* \mathcal{F}_\mathbb{R})$ has an algebraically isolated singularity at the origin and the algebraic multiplicity satisfies $\nu_0(\mathcal{F}_\mathbb{R}) = \nu_0 (i_\mathbb{R}^*\mathcal{F}_\mathbb{R})$. A germ of foliation $\mathcal{F}_\mathbb{R}$ is of \emph {topological real generalized surface} type if, for all immersion $i_\mathbb{R}:(\mathbb{R}^2,0) \hookrightarrow (\mathbb {R}^3,0)$ transversal to it, the foliation $i_\mathbb{R}^*\mathcal{F}_\mathbb{R}$ has no real topological saddle-nodes in the process of reduction of singularities. As an application of our main result for $(\mathbb{R}^2,0)$, we show that \emph {a germ of topological real generalized surface foliation in $(\mathbb{R}^3,0)$ having even algebraic multiplicity, or such that there is at least one transverse immersion for which the Milnor number $\mu_0(i_\mathbb{R}^*\mathcal{F}_\mathbb{R})$ is even, has at least one formal separatrix}.