A Hölder continuous vector field tangent to many foliations

Abstract

We construct an example of a Hölder continuous vector field on the plane which is tangent to all foliations in a continuous family of pairwise distinct $C^1$ foliations. Given any $1\le r<\infty ,$ the construction can be done in such a way that each leaf of each foliation is the graph of a $C^r$ function from $\mathbb{R}$ to $\mathbb{R}.$ We also show the existence of a continuous vector field $X$ on ${\mathbb{R}}^2$ and two foliations $\mathcal{F}$ and $\mathcal{G}$ on ${\mathbb{R}}^2$ each tangent to $X$ with a dense subset $\mathcal{E}$ of ${\mathbb{R}}^2$ such that at every point $x\in \mathcal{E}$ the leaves $F_x$ and $G_x$ of the foliation $\mathcal{F}$ and $\mathcal{G}$ through $x$ are topologically transverse.