Campo de vetores Hölder contínuo tangente a várias folheações

Abstract

The objective of this work is to bring the positive answer presented by Bonatti and Franks to the question proposed by Wilkinson about the existence or not of a continuous non-singular vector field of $\mathbb{R}^2$ tangent to more than one foliation. Thus, as the central bundles are always continuous Hölder, Bonatti and Franks provide an example of a continuous vector field Hölder in $ \mathbb{R}^2$ with the property that there is a family of foliations of class $C^1$ distinct and tangent to each foliation, with each leaf of each foliation being the graph of a function of class $C^r$ from $\mathbb{R}$ to $\mathbb{R}$.