A H\"older continuous vector field tangent to many foliations

TL;DR

This paper constructs a H"older continuous vector field on the plane tangent to a continuous family of distinct $C^1$ foliations, with leaves that are graphs of $C^r$ functions, and demonstrates complex tangency and transversality properties.

Contribution

It provides explicit examples of vector fields tangent to many foliations with leaves of arbitrary smoothness, and shows the existence of vector fields tangent to two foliations with dense topologically transverse leaves.

Findings

Constructed a H"older continuous vector field tangent to all foliations in a continuous family.

Showed existence of a vector field tangent to two foliations with dense topologically transverse leaves.

Demonstrated that leaves can be graphs of $C^r$ functions for any $1 \\le r < \\infty$.

Abstract

We construct an example of a H\"older 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 $\R$ to $\R.$ We also show the existence of a continuous vector field $X$ on $\R^2$ and two foliations $\cal{F}$ and $\cal{G}$ on $\R^2$ each tangent to $X$ with a dense subset $\cal E$ of $\R^2$ such that at every point $x\in \cal E$ the leaves $F_x$ and $G_x$ of the foliation $\cal{F}$ and $\cal{G}$ through $x$ are topologically transverse.