Preprint C91/2009
Ciclos Limites Projetivos e Aplicações Computacionais à Dinâmica Complexa
Evilson Vieira
Keywords: holomorphic foliations | projective limit cycles | 16${}^\textrm{th}$ Hilbert's Problem
For a holomorphic foliation $\mathcal{F}$ in $\mathbb{P}^2_\mathbb{C}$ with isolated singularities and defined over $\mathbb{R}$ we have also the real foliation $\mathcal{F}_\mathbb{R}$ in $\mathbb{P}^2_\mathbb{R}$ for which the leaves are obtained by the intersection of the leaves of $\mathcal{F}$ with $\mathbb{P}^2_\mathbb{R}$. If $\delta$ is a cycle of $\mathcal{F}_\mathbb{R}$ we have two possibilities: $\delta$ is homotopic to a point in $\mathbb{P}^2_\mathbb{R}$ or $\delta$ represents the generator of the fundamental group of $\mathbb{P}^2_\mathbb{R}$. In the first case $\delta$ is called an affine cycle and in the last case $\delta$ is called a projective cycle. In this work we study projective cycles in $\mathbb{P}^2_\mathbb{R}$. Our inspiring example is the Jouanolou foliation of odd degree which has a projective limit cycle. We prove that only odd degree foliations may have projective cycle and odd degree foliations with exactly one real singularity, and this singularity is non-degenerated, has a projective cycle. We also prove that if a generic Hamiltonian foliation has a projective cycle the cycles near to projective cycle are vanishing cycles and prove that after a perturbation of a generic Hamiltonian foliation with a projective cycle, we have a projective limit cycle if, and only if, the perturbation is not Hamiltonian. In addition, we present a graphical platform whose objective is auxiliary the study in Complex Dynamics.